If the Yorkists lose, Percy's army isn't going anywhere; Henry's priorities will be to secure the king, and he certainly has the troops to defeat Percy before he can meet up with reinforcements. Most likely at this point, if Percy has Richard, he turns him over to Henry in exchange for being forgiven for opposing him. If he doesn't have Richard, he still makes whatever peace he can; he's not in a good position to continue resistance, and he's just seen Richard's main army be routed. The Tudors will absolutely pursue Percy's forces if they continue to resist; their commanders are certainly capable enough for that, and if Percy finds himself outnumbered and surrounded, he has little choice.
Certainly Burgundy, Ireland, etc. can raise additional troops if given time, but Richard needs them now if he's to avoid becoming a hunted fugitive.
Defeated armies in the Wars of the Roses rarely retreated in good order; if Richard is truly defeated (which I take to be the POD), he's not reforming his army anytime soon. OTL, quite a few escaped due to Richard's death-or-glory charge leaving them behind; if he loses but still survives, the POD essentially requires him not to launch said charge, in which case his army is going to have a much more difficult time disengaging safely. If nothing else, Henry's surviving army is (by assumption, given that he won ITTL) significantly larger than what's left of Richard's, and with a significant victory under his wing, is likely to attract more supporters who see the way the wind is shifting, while Richard's army will be severely demoralized. Henry already has the most significant nobles either on his side (the Stanleys) or dead/defeated (Percy, Howard); Richard needs time to build an army to replace the one he mostly lost. At best, he escapes to the north with Percy and whatever troops he can salvage, but doing so essentially concedes the more populous south, and automatically confers significant legitimacy to Henry.
As for the question of whether he's a pretender or not, that's mostly semantics, but if he's been badly defeated and driven away from London, he's certainly not the uncontested symbol of legitimacy.