See also my 'Alternative History of Britain: The Hundred Years War' (Tim Venning, pub. Pen and Sword, 2013) page 109 and ibid, 'The Wars of the Roses' (Pen and Sword 2013) p. 9, where I briefly ,mention the possibilities arising from either Duke John of Bedford (d 1435) or Duke Humphrey of Gloucester (d 1447) having legitimate sons. As in this article, I also infer that Humphrey's heirs would have been in a weaker position re: the English succession, as his first marriage, to Jacqueline of Holland, took place in dubious circumstances affected by her disputedly legal divorce from her first husband - a close relative and ally of England's ally Duke Philip of Burgundy - which was later invalidated by the Pope under Burgundian pressure. This would have bastardised any son of Humphrey's by her, and his second marriage , to Eleanor Cobham, was disliked by many nobles and was open to attack politically by his enemies as she had been a low-born lady in waiting and so was not of the usual 'acceptable' elite class for a great noble/prince's wife; it was thus easier for his enemies to get this marriage invalidated too.
So if Humphrey had been insisting in the 1440s that after himself his son , by either woman, was his heir to all his lands (which in OTL went to the impoverished and financially needy King and the allegedly extravagant Queen as he had no legal heir) and thus to the Crown too, this would be open to attack whether or not he was arrested and ruined and/or died in 1447. The Beauforts as well as York's faction could have targeted Humphrey's heir. A son of Bedford by his OTl wife Anne of Burgundy would presumably have been born in the mid-late 1420s and not been adult enough to succeed Bedford in his role as regent of/ commander in France in 1435, meaning that that role went to Humphrey and then to York as in OTL, and even in the early 1440s a young and untried son of Bedford's would have been at a disadvantage even compared to the more experienced John and Edmund Beaufort. But possibly the King and Suffolk's faction could have bought him off from joining their critics after York was recalled and sidelined from power and Humphrey was destroyed,in 1447, by giving him a command in Normandy - though by that date the lack of men or money would have meant that he could not do much. Had he put up a more respectably aggressive defence there in 1449-50 than Edmund Beaufort did in real life, this could give him support after the end of English Normandy in 1450 and make him a potential rival to York - if he was so inclined, eg by bitterness at the Beauforts or their ally Suffolk not giving him enough men and money to hold out longer. Or anger at them being too keen to hand over Maine to France on easy terms but keep this shabby deal secret from the English public in 1445-8?
Bedford married Anne in June 1423; a son of his by his second wife Jacquetta of St Pol (by her next marriage the mother of Edward IV's Queen, Elizabeth Woodville, and about a dozen other children), married April 1433, would have been in an even poorer position until c. 1450 as too young to take part in war or politics. After 1453, when Henry VI had his first outbreak of a catatonic stupor, they would have been in a good position to challenge York as the reversionary heir, or once his son Edward was born later in 1453 as regent; and as Henry's death would not mean York becoming king, this might have eased the attitude of fear and hatred to York by the Queen and her (Beaufort-led) faction and/or stopped anti-Beaufort nobles all gravitating to York's side. Also, if the crises had developed as in OTL and led to York seizing power as 'regent' in 1455 (as older and more experienced than Bedford's son?) and then being attacked and forced to flee for his life to Ireland by the Queen in 1459, York would not have been able to claim the throne on his return in 1460. A different dynamic to the so-called Wars of the Roses and a three-way fight between the Queen's party, 'the second duke of Bedford', and York?
Bedford's son would have been much stronger in political terms, or been a senior military commander in France if competent enough, had Bedford (born 1389 and so a late 34 at marriage) married earlier - eg if his father Henry IV had married him off in his international schemes of the late 1400s, and/or his intervention in France to back up the 'Armagnac' faction against the Burgundians in 1411-12. At this point, with the future Henry V the heir but at odds with his father and favouring Burgundy instead, Bedford was however not the next heir, to whom the King would turn if at odds with his eldest son - this was Henry IV's second son Thomas, Duke of Clarence (b 1388, k 1421). The King could still have arranged a good, probably French marriage to an heiress to lure Bedford to support him. Clarence, killed by the French and Scots in battle at Bauge in March 1421 in OTL, left no children by his 1411 marriage to the elder John Beaufort (the King's illegitimate half-brother)'s widow Margaret Holland; but as said, a son of this marriage (born c. 1412-16?) would have been in a good position to be a senior commander in France after 1435 and , as adult, would also be the next heir to Henry VI ahead of Humphrey so this might make Humphrey less of a threat to the King and reduce the chances of his arrest in 1447. Humphrey was however naturally truculent and quarrelsome, as seen by his long feud with Bishop Henry Beaufort (the next brother of the elder John Beaufort), and he even undermined his brother Bedford in the mid-1420s by marrying Jacqueline of Holland and so annoying Bedford's vital ally Philip of Burgundy.
The possibility also arises of a son of Bedford (if born before c. 1418) or of Clarence succeeding as full viceroy / governor of Normandy and the rest of 'English France' (with Paris to 1436) after Bedford died in 1435, though only as regent - ie with sovereign powers - until Henry VI's majority in 1437. Then if they were not given adequate troops or money to fund them , the English position would collapse in the mid-later 1440s as in OTL, once Charles VII had got a proper army organised and funded and had put down his feuding nobles in the major 1440 revolt; or if 'Bedford II' was less aggressive and more cautious the abortive peace-conference of 1439 at Gravelines could have established a truce that kept the status quo for a few more years but was later broken by a reinvigorated France. (As the truce proposals included France insisting that any full peace would require Henry VI to abdicate as king of France and to be a French vassal duke of Normandy and Aquitaine, and H VI always refused this, then no long term peace was viable.) So could 'Bedford II' be evicted from France by or soon after 1450, be furious at the lack of support sent by Henry VI - mainly due to incompetent planning and systemic over-spending plus a corrupt court - and turn on his cousin the King in a different sort of civil war to OTL? And end up as 'King Henry VII' or 'King John II' with the - unreliable - backing of York, provided that he had no son to succeed him and he gave this heirship to York?
Bedford married Anne in June 1423; a son of his by his second wife Jacquetta of St Pol (by her next marriage the mother of Edward IV's Queen, Elizabeth Woodville, and about a dozen other children), married April 1433, would have been in an even poorer position until c. 1450 as too young to take part in war or politics. After 1453, when Henry VI had his first outbreak of a catatonic stupor, they would have been in a good position to challenge York as the reversionary heir, or once his son Edward was born later in 1453 as regent; and as Henry's death would not mean York becoming king, this might have eased the attitude of fear and hatred to York by the Queen and her (Beaufort-led) faction and/or stopped anti-Beaufort nobles all gravitating to York's side.
The relative youth of a son of Jacquetta when everything is kicking off would be interesting. Even in 1453, he'd still only be a teenager- can he credibly challenge York for the Regency? Or would Margaret and the Beauforts see him as young and malleable and exalt him to the Protectorship to exclude the dangerous York?
To what extent do you think not being heir- never being heir- would affect York's behaviour in the early 1450s? I suppose this is somewhat dependent on how you interpret York's actions on the ''completely selfless champion of good and responsible government'' to ''self-interested, ambitious, magnate'' spectrum?
There'd also be a bunch of nitty gritty stuff to get into if one were exploring things further- is Bedford married? Is his father-in-law or brother-in-law be a reliable ally (...against York, against the King if it comes to that)? Where are Bedford's lands? Does he have quarrels or feuds with any neighbouring magnates? How does that effect possible alignments? Etc.
A messy three-faction struggle between York, Bedford, and Margaret/the Beauforts would be fascinating.
If a son of Bedford, born either c. 1424-8 (son by Anne of Burgundy) or c. 1434 (son by Jacquetta), is available to be regent in 1453 (Henry in a stupor and has no son yet) or 1455 (Henry defeated in battle of St Albans and Edmund Beaufort killed there by York, so York leads new regime but by now Henry has a son aged 18 months), 'Bedford Jr' is the closest adult male living relative to the current King. That legal argument of 'next adult heir becomes regent or at least is in personal charge of an under-age or sick king' made the future Henry IV, as nearest adult cousin, regent for the arrested Richard II - and soon enabled him to take his throne - in 1399, and left out the nearest non-adult cousin to Richard, Edmund Mortimer, who was genealogically closer but was aged 8. In 1327, when Edward II was deposed by his wife Isabella and their 14-year-old son Edward III succeeded, as the latter was not yet adult he was put in the charge of his nearest adult male cousin, Earl Henry of Lancaster, but a different and politically more strong figure - Isabella - seized the regency.
So in a legalistic age where the literate and legally expert Churchmen plus lay judges would usually be called on by the nobles to provide legal precedents to back up a course of action, either 'Bedford jr' could be regent and in charge of Henry VI (plus Henry's infant son) in 1455, or the two roles be separated. if York has led the 1455 revolt as in OTL - he alleged that he was left with no option to save his faction from a purge by the Queen and the Beauforts who had called a 'packed' Parliament ready to prosecute them- but Bedford jr was not involved in the revolt as the Beauforts or Queen paid him off, then York could seize the regency as older and more experienced than a young man of around 20 with no major record of governance or leadership in war. (The French have just taken Normandy 1450 and Bordeaux 1453 and the nation is expecting a campaign to retake them.)
York could still lead a successful faction to challenge the Queen/ Beaufort faction after the disaster in Normandy in 1450 or the fall of Bordeaux in 1453, if Bedford jr is too young to have commanded in the war and so built up a good record and assembled some powerful noble proteges; York would be the 'last good commander who achieved anything in Normandy, in the early 1440s' as in OTL, and would be claiming that the Beauforts and the late Duke of Suffolk (their ally, lynched in a revolt in 1450) let him down and betrayed the English cause by their lack of support , their embezzlement of money that should have been used on troops, and their treasonous sell-out to France in 1445-8 by promising to hand back Maine (S of Normandy) in return for a failed truce and the King marrying the French King's niece - ie the present Queen. Margaret of Anjou. York is the natural claimant to a 'patriotic hard man out to purge the corrupt court' role, as used successfully by Simon de Montfort against Henry III in 1258 and 1264 and by Thomas, earl of Lancaster (also the King's nearest male cousin) against Edward II in 1308, 1312 and 1322 - this is a role that nobles regularly rally to, if excluded from power and jobs at court. But we could well end up with a sidelined Bedford jr then gravitating back to the Queen in the late 1450s and helping her to exile York in 1459 - and a struggle over who succeeds Henry in 1460-1. Logically, if Bedford jr is born c. 1434 as Jacquetta' s son, then he could be bought off by one faction or other in the early-mid 1450s by giving him an heiress from their leading families. York has daughters to offer, logically Anne (the eldest) born in 1439 as the nearest in age; or there are various female Beaufort kin. If the 'Bedford jr' role is taken by Thomas of Clarence (k 1421)'s son by Margaret Holland, though, he is older - born by 1421 - and no doubt married already, and he is already kin to the Beauforts as his mother previously married Edmund B's uncle John . So he is a natural Beaufort ally, unless he has been sidelined by Suffolk and the Queen after 1447 as too friendly to Duke Humphrey or critical of the mid-1440s truce negotiations. In that case, do York and his wife's brother Richard Neville (the Kingmaker's father) buy his allegiance with a Neville bride?
if Bedford's father lasts a few more years after 1435, then Jacquetta either does not marry her next husband Sir Richard Woodville (1436? in OTL) at all or does so in the 1440s; so her eldest daughter Elizabeth Woodville , if born at all, may not be married off to Sir John Grey and widowed when he is killed in the 1461 civil war. So she is not available to marry Edward IV in 1464; he could marry the French candidate who his cousin Warwick 'the Kingmaker' wanted him to marry, Bona of Savoy, or even the Spanish candidate, Isabella the sister of the then king Henry IV of Castile. So we could get a possible son for Edward and Bona/ Isabella, born before the OTL date of Edward V's birth 1470, meaning that there is a King of 16 or 17 when Edward IV dies in April 1483 - no need for a regency then and no allegedly greedy Queen Elizabeth 's Woodville faction in control , so no excuse for a coup by E IV's brother Richard III. No Tudors on the throne. Or we have a 'Queen Isabella of England', so no union of Castile (Isabella) and Aragon (Ferdinand) but Henry IV's daughter Juana keeps Castile separate after 1474. All sorts of butterflies for world history...