Well if Exeter was to parlay the fact that he had a son or grandson (via Anne) and he had the most senior claim to the throne to become leader of the Lancastrian exiles over the younger Henry Tudor, the next important thing to getting enough men to challenge Richard III.
What helped Tudor at Bosworth was that Richard III's supporters were abandoning him, most famously Tudor's step-father Lord Stanley, or decided to see how the tide of battle was going before joining. Also Tudor while nominally in command of his army, it was actually Tudor's uncle Jasper and the Earl of Oxford as the field commanders. While Exeter would have the services of Oxford, but would those supporting the Tudor/Beaufort Lancastrian claim join Exeter?
Also how would Exeter's grandson, who's paternal grandmother was Elizabeth Woodville, as the heir-apparent play in England since some nobles didn't like the Woodville faction?
As for Yorkist princesses, my two cents would be this. Depending on the age of Exeter's eldest grandson (and thus heir-apparent), if Exeter were to win "Bosworth" and become Henry VII. He would look secure the future of his dynasty and would marry Elizabeth to his grandson the future king to ultimately unite the two claims to the throne.
As for Henry Tudor's future in a Exeter-as-King TL, if he supported Exeter in battle he would probably be given his father's earldom and be given the task of insuring Wales came into line with the new regime. If Tudor hung back while Exeter won the crown, he would eventually be given his inheritance after his mother pleaded on his behalf but because he didn't help the Lancastrian cause he would be persona non grata. In either case, Exeter would ensure Tudor doesn't get any closer to the throne than he already is most likely having Tudor marry a daughter of the Nevilles of Westmoreland or a Percy.
Finally, what if Exeter were to die while defeating Richard III? Well, Exeter's son-in-law Thomas Grey would most likely become King via his wife, Anne Holland. Thomas' son (Exeter's grandson and heir-apparent) would be a minor and while some would not want to put Grey on the throne, it was either Thomas or a minor with Thomas as regent and after Henry VI and what happened to Edward V it would be seen as preferable to let Grey be crowned.