Doesn't a living Edward of Middleham make Richard's reign marginally more secure? Richard might also be not so desperate to remarry if he has a living heir and hence avoid the damaging rumours about Elizabeth of York (or, without the grief of her son's death, Anne Neville might live a bit longer). If he has a son to worry about Richard might also be less inclined to do his reckless, all or nothing, charge at Bosworth.
That said, if things do go as OTL, I imagine the likely place Edward will flee to is Burgundy, to seek shelter and eventual support from his aunt Margaret. It doesn't really make sense for him to go to France, considering the French just supported Henry Tudor.
Of course, precisely what happens in the immediate aftermath of Bosworth depends on whose custody Edward is left in- and whether his guardian decides to launch some immediate ill-considered anti-Tudor uprising with Edward as a figurehead, or whether they're okay with taking him to relative safety on the continent and letting him grow up a bit.
If they do have a unifying figure to rally around it's possible some of Richard's loyalists gravitate towards Edward (wherever that may be) rather than getting themselves killed in haphazard rebellions like the Stafford and Lovell Rebellion.
If Edward does end up in Burgundy, it's possible he gets caught up in Lincoln's enterprise- Edward would be a much better figurehead than Lambert Simnel (impersonating the Earl of Warwick despite the fact that Henry Tudor had the real Earl of Warwick in his custody). Wherein Edward will end up dead or imprisoned after something analogous to the Battle of Stoke, unless Edward's presence somehow swings things in the rebels favour (which seems doubtful) or they otherwise get lucky.
Alternatively, given they have a bona fide claimant here some of the Yorkists might be inclined to take the slow burn approach (diplomatic recognition, politicking, and all that) rather than throwing Edward's life away on something as risky as the Simnel Rebellion was IOTL. Or, you have some compromise where Lincoln et al. go off to raise rebellion in Edward's name whilst Edward himself remains safely ensconced on the continent.
Long term, Edward can probably count on support from aunt Margaret (though how overt this support is will depend on diplomatic realities), but'll probably end up with a Warbeck-esque career of wandering around looking for support and being a diplomatic bargaining chip. Whether he's anymore successful than Warbeck will depending on how much advantage can be gained from his less dubious past, and to what extent he can stir up support in England.