in practical terms, as England lacked a large army at this date - apart from a few Royal Court residential regiments and the Calais garrison it had to raise levies of 'volunteers' for specific and fairly short time-periods from contractors in each county, usually great noblemen - it could not keep an army in Scotland for more than a few months, except for a few garrisons near the Border that could be kept supplied over the winter. If even a powerful Regent with royal legal authority and the backing of all of his Council (Seymour had purged his conservative , Catholic rivals from it before or soon after Henry VIII died and may have fiddled with his will) ordered his army to stay in Scotland over the winter and broke the troops' contract for a shorter length of service , he would face major trouble and desertions
This would revive his aristocratic foes' hopes of using this to stage a revolt and replace him, as happened in OTL in autumn 1549 after his contentious religious reforms and ruthless Protestantisation of the Church (and seizure of the Catholic chantries and their property) sparked off both religious and anti-landowner rural social risings. (Plus the final ant-Anglicization, anti-English language, pro-Catholic 'Western Rising' in Cornwall in 1549.) To secure his position at home and keep enough troops there - and hang onto Henry VIII's acquisition of Boulogne, taken by English troops in the invasion of 1544 - he could not keep many troops in Scotland; in OTL with a largely hostile Scots elite he could only hang onto Haddington (near the coast so easier to supply) in Lothian in 1547-8 and even with a more subdued Scots elite he's going to be lucky to be able to risk keeping enough troops in Edinburgh, presumably plus its port of Leith and the major towns and castles in the locality, to overawe resistance. Taking and holding somewhere crucial like Stirling , lynchpin of Edward I's invasion and occupation force in the 1300s, is possible with his artillery but holding onto it as well is dubious, and occupying anywhere North of that for more than a few months is impossible with his resources - he's going to need to work with the majority of the Scots elite and not impose his will too obviously or they'll turn to France.
Wanting Boulogne back and unwilling to sign peace, the warlike and aggressively Catholic Henri II is going to want to distract and weaken Seymour's govt so he will be eager to meddle in Scotland; and even if Seymour has the luck to get his hands on Mary Queen of Scots in 1547 (paying allies in the nobility to kidnap her and then carrying her off to London by sea for a forcible engagement to Edward VI - Mary was only 5 in Dec 1547 and Edward was 10) there's the question of angry Catholic nobles, foes of Henry VIII's interference in 1543-7, linking up to France in retaliation. The pro-Henry and pro-Protestant nobles in 1542-7 were led by Matthew Stewart, earl of Lennox and head of the junior royal Stewart line with the next 'male descent' line of the royal family to Mary (who has no uncles or close Scots Stewart cousins left alive) , so they would be linked to Seymour - Lennox was married to Henry VIII's niece Margaret Douglas, linking him up to the powerful Douglas dynasty who dominated SW Scotland (Margaret's father, widower of H VIII's sister, was head of the main Douglas line), so they will back Seymour if paid enough and the Protestant Seymour can win over some nobles by staging an earlier version of the OTL Scots Reformation of 1560 and confiscating abbeys and their estates to sell them to nobles in return for support. The main cheerleader of the Reformation, preacher John Knox, was available to support this and had already been backing Protestant rebels when they seized St Andrews and murdered Catholic Cardinal Beaton in 1546 - though he had then been captured by the Scots govt's French allies and shipped off to France as a convict so he would have had to retrieved and ransomed first. Staging an earlier Reformation and secularising the Church will help bring lots of nobles and lower landholders to back the English if it is officially led by Seymour's Scots allies in an 'inclusive' manner by a supposedly autonomous lay and Church assembly, not imposed at gunpoint as an 'alien' policy - but that will then drive the hardline Catholic nobles who oppose any reform, led by the Hamiltons (who also had a claim to the throne, as Mary's closest relatives in the female line of Stewart descent) , to call in French aid.
So the realities of the lay and religious divides in Scotland in the late 1540s are going to end up in civil war, rebellion, and French intervention at some point , sooner rather than later unless Seymour is more subtle and less arrogant than he was in reality - and his lack of will or ability to keep all his regency Councillors onside and happy in England in real 1547-9, and his hurrying through the Reformation there, led to revolts and then his deposition in 1549 so he is not a clever political operator. (To make matters worse, his jealous younger brother Thomas Seymour is trying to undermine him and will end up executed by him so even the Seymours are not united. See the current TV show 'Becoming Elizabeth' - which is not as popular as 'Wolf Hall' etc so we seem to be getting to 'saturation point for the Tudors on TV!). To steer his way through this mess and keep the French from landing troops to destabilise Scotland or at least sending money and arms to the Hamiltons seems very unlikely , at least in looking 4 to 5 years ahead - and unless Seymour avoids the 1549 risings in England he's going to need to pull troops back then and leave an open field for the French. Perhaps he can stave off a 1549 revolt if he does not stage so drastic or quick a Reformation in England yet, with no move to a fully Protestant prayer-book, or he has to bring more of the dubious and not pro-Protestant English nobles with him instead of having full royal powers as Regent; a better balanced Council could rein him in on religion yet still give him full support to invade Scotland. It is still all likely to go wrong unless he is both careful and lucky, given France's strength and Henri's Catholicism.
But if Mary QS is in England from 1547 and is a hostage at the Court, bringing her up as a Protestant, that has interesting possibilities; when Edward dies in 1553 she is still only nearly 11, ie too young to marry him and have children, and the new Queen Mary Tudor is a foe of France and will not hand her over to Henri II or to her Catholic mother Marie de Guise, a Frenchwoman and Henri's ally. Will Mary Tudor marry her cousin MQS off to some minor Habsburg with the help of her fiance, King Philip II of Spain? And will Spanish troops then help to reverse the Reformation with the help of the Hamiltons and Gordons, and keep Scotland Catholic?
