There's an intriguing question-mark over the real attitude of Buckingham to both Henry Tudor and Richard III, and whether he was after the throne for himself - and thus only pretending to support Tudor in order to get the Woodville-Edward IV's ex-Household 'loyalists' within the autumn 1483 conspiracy to help him. The key figure here seems to have been Bishop John Morton of Ely, a devious former Lancastrian aide to Margaret of Anjou who had joined the Yorkists and got a bishopric and admin jobs out of Edward IV but who when Edward died in 1483 got arrested by Richard for his part in the supposed 'plot' by Lord Hastings, Elizabeth Woodville, and Jane Shore in June. Richard put him in Buckingham's custody at Brecon, to get him out of London without annoying the Church by a more formal imprisonment or sacking/ executing him, and Morton apparently 'turned' Buckingham to betraying Richard and linking up to the Woodvilles - or B decided to use him as the contact to them, having already decided to betray Richard. After the failure of B's rebellion he fled to France and joined Tudor - and returned to England with him in 1485, ending up as chancellor and Archbishop of Canterbury.
B had been given virtual viceroy powers in Wales, where he was the lord of the Brecon/ Radnor lands,by Richard,
so this was rank ingratitude as Richard fumed - but was he annoyed at not getting more of the inheritance of his ancestress Eleanor Bohun, co-heiresss in the late C14th of the Brecon/Builth lordship, from Richard? The lands of the Bohun dynasty, earls of Hereford, had been divided between the last earl's daughters Eleanor, married to Buckingham's ancestor Duke Thomas of Gloucester who was ex. 1397, and her sister Mary, married to King Henry IV, so Eleanor's daughter and her husband, Lord Stafford,and their son, B's grandfather Duke Humphrey of Buckingham, only got half of them; Henry IV and his heirs Henry V and then Henry VI got the other half. As of 1483 the 'Henry IV half' was in the crown's hands, and B seemingly wanted Richard to give all this to him. This is usually cited by most historians as why Buckingham turned on Richard, out of greed; but some allege that he was using Richard as a route to the crown for himself , and that he then used Morton too. But was he really so clever and devious? There is no indication in the sources for definite of his political skills; but he seems to have been arrogant and greedy and possibly also a snob, if the story about him being annoyed at Edward IV forcing him to marry a 'low-born' Woodville is true. Edward IV never employed him , which may mean that E did not trust him or that he was seen as no great asset in politics pre-1483.
Thomas More in his c. 1510 (abandoned) biography of Richard, the main source of Shakespeare's play, alleged that B was horrified when he found out about the murder of the Princes - his wife's nephews - and so turned on 'serial killer' Richard in case he was next on the list;but was this a whitewash by TM to please Buckingham's son Edward, who by this time was restored as Duke and was a senior figure at H VIII's court? (Henry later executed Edward for a 'plot', in 1521.)
There is also a question over the finale of B's life, when he was captured and taken to Richard at Salisbury in early Oct 1483; he apparently asked Richard for a personal one-to-one interview as he had secret info to tell him, but Richard refused it and had him executed (in Salisbury marketplace) instead. One Tudor-era story (from B's son Edward or his surviving household retainers?) had it that B created this appeal as an excuse to get close to Richard, and intended to kill him at the interview; if he had done so, and then been killed by R's guards, the reign of Richard would have come to an abrupt end earlier than in OTL. Legally, the next heir would then be Richard's son Edward of Middleham, born c. 1474 and in poor health as he was never allowed to come to unhealthy London in his father's reign and he died in April 1484. (His mother Anne Neville probably had TB.) So who would be regent - Anne, or Richard's senior adviser John Howard, the new Duke of Norfolk? And if 'King Edward VI' dies in 1484 on schedule, who is King then - Richard III's OTL heir as of 1485, his sister's son Johh de la Pole?
An alternative version has it that the 'secret info' was the whereabouts of the Princes in the Tower, who had disappeared from the Tower but not been killed by Richard - Buckingham had smuggled one or both of them to safety abroad so he could use them as his candidates in a later rebellion. The Tudor era pretender Perkin Warbeck, alleging himself to be the younger Prince, said that a 'great lord' - unnamed - had smuggled him abroad but told him to live undercover and not name his rescuer. So was this B, as a Flemish source c.1500 claimed? I go into all this in my forthcoming book on 'Medieval Royal Mysteries', to be pub by Pen and Sword, date unclear yet but probably 2021/22 - my King Arthur book and probably my Anglo-Saxon Mysteries book will come first. Text already in place.