I think, sooner or later, Edward finds himself back in England at the head of an army. He's a prince of royal blood, and is going to be head of the Lancastrian side of the family. Sooner or later, he probably decides to set sail for the British Isles.
Probably, but a lot depends on whether the Battle of Tewksbury or something equivalent happens in TTL. If it is fought, the Beauforts and other Lancastrian leaders probably die as in OTL, but Edward of Westminster’s courage will be questioned and he may have difficulty finding support in the future.
If the battle isn’t fought, then I’d expect the Lancastrian leaders to withdraw to France and try again some time in the next few years. Their performance against Edward IV was less than stellar in OTL, but they might get lucky.
Either way, Henry VI probably lives longer than in OTL. He’s still useful as a puppet/bargaining chip for the Yorkists and he’s not going pine away if his son hasn’t died. OTOH, he did outlive his father and grandfather, so he may just die on schedule.
Edward of Westminster’s marriage to Anne Neville may be annulled. It was only done to cement an alliance with her father, Warwick, and he and his brother are dead before Tewksbury. If not, she doesn’t seem to have been very fertile.
Henry Tudor does not become the primary Lancastrian claimant to the throne. Edward of Westminster, as well as Edmund and John Beaufort (if they survive in TTL) have better claims and will likely be producing heirs instead of dying without issue.
Butterflies also affect Edward IV’s children, since Richard of Shrewsbury and his younger sisters haven’t been conceived yet in 1471. Most of his brother George of Clarence’s children (and all that survived to adulthood) were born after 1471. Butterflies will also affect Richard of Glouchester’s children, even if he somehow still marries Anne Neville in TTL.