Assuming Henry Holland Duke of Exeter still dies (this occurred in late 1475) and Tudor can therefor still claim to be the last 'Lancastrian' claimant (legally the Lancastrian claim passed to some Iberian royal families).
With Tudor having a decade in England what would his relations with his his Stanley step-family be like? Are they more likely to support him? Does a closer relationship between Thomas Stanley and a potential claimant mean Richard III is more wary of the Stanleys/excludes them from gov/otherwise neutralise them (IOTL he gave custody of Margaret to her husband after the Buckingham rebellion, which allowed her to keep plotting).
If things go relatively recognisably and Buckingham still rebels against Richard III that'll be different- he can't switch his support to some 'king over the sea' when he finds out the princes are supposedly dead, maybe claims the throne himself.
Now, not having an obvious figure to congregate around will damage the coherence of any Woodville-party exiles, and without someone hanging about their court the French probably won't take as much interest in England.
In addition to not getting French support a non-exiled Henry doesn't meet up with the Earl of Oxford and might be separated from his uncle Jasper Earl Pembroke. This hurts his chances but a shorter exile probably makes him a happier, less-bitter/paranoid person.
Now who would he marry? There was talk during the later years of Edward's reign of marrying him to Elizabeth of York if he returned to England, but I can't see an impoverished exile of dual bastard ancestry marrying a princess. There was talk of him marrying Maud Herbert, daughter of his warden William Herbert. But there are probably other matches out there. If Henry has any children in the intervening decade that also affects any ability he might have to take the throne- either by bargaining off marriage alliances or having his family captured and held hostage.
Finally, and this might just be my cynicism, but is there any chance that Edward's plan was to lure him back so he could be eliminated as a threat via poisoning or trumped up treason charges or something? I mean, he supposedly had Exeter murdered, and the Woodvilles may have orchestrated Clarence's execution because they saw him as a potential threat (though George had admittedly done enough by himself to get executed).