This basically has to happen before Henry VII gets his 'heir and a spare' in 1491, unless the POD is that he's infertile or something. This rules out the Duke of Buckingham, the Countess of Salisbury's sons and the Earls of Devon on age grounds, so the only adult men who would claim the throne would be the Earl of Warwick (still locked up in the Tower) or, far more realistically, the de la Poles. Furthermore, Edmund de la Pole seems to have been a bit of an idiot, and OTL didn't dare rebel: he lived in England until 1501 without too much suspicion coming his way.
This leaves the whole John de la Pole/Lambert Simnel/Stoke Field thing as the last best chance the Yorkists had (I exclude the Lancastrians because they were all either A) foreign monarchs; or B) loyal to the Tudors, eg Ralph Neville, 3rd Earl of Westmorland). Stoke Field was weighted heavily in favour of Henry VII - he had slightly more men, had more experienced commanders, and was less dependent on foreign mercenaries. But say the Battle goes the other way, which is quite plausible. Arthur, Prince of Wales, has already been born, so even if Henry VII dies on the battlefield, his supporters have got their very own 'King Arthur' to hawk in every marketplace. If they are then defeated and Arthur goes abroad, or even dies, he will be comparable to King Sebastian of Portugal in terms of command over the imagination.
If Henry VII survives, well, he already has the loyalty of most nobles, and everyone is pissed that the War of the Roses simply WILL NOT END. And who has Simnel got on his side? Lincoln (an untried courtier), some Irish blokes and a German mercenary captain. That isn't the kind of line-up that inspires support from Joe Peasant. Also, Lincoln's motives are unclear. If he puts Simnel on the throne, will he do a Richard III on the real Warwick? What's in it for him if either Real!Warwick or Fake!Warwick get the throne, anyway? I suspect he'd want the throne for himself - two of his brothers certainly did, when there were arguably better claimants knocking about.
So yeah, I'd say you'd get King John II, with two or three child murders to his name and an inauspicious name. After him, either a son, a brother, or one of the other claimants riding a wave of righteous indignation in 1490 or so.