Tudors of Britain

In OTL Henry Tudor became King of England following the death of Richard III at the Battle of Bosworth, and founded the Tudor dynasty, with one of his daughters, Margaret, marrying James IV of Scotland, leading to the eventual union of the two crowns and of the two kingdoms thereafter.

What other possibilities could there be?

This is a partial list of possibilities: feel free to add to the list or develop possibilities (in which case add a link)

Henry VII
Richard III arranges for Elizabeth, the eldest daughter of Edward IV, to marry someone else (possibly into the Scottish ruling dynasty), weakening Henry Tudor's position.

Richard is merely defeated at the Battle of Bosworth and goes into exile, posing a threat to the new regime.

Richard dies, but various of the Yorkist claimants and associates (Francis Lord Lovell, Lambert Simnel, Perkin Warbeck) manage to pose a stronger threat.

Henry VII takes up Christopher Columbus's request for patronage: England and Portugal become the dominant powers in the Americas (but the Spanish Pope Alexander VI is neutral towards the two sides, unlike OTL)

The succession
Margaret is married into another royal family.

Henry VII's elder son Arthur survives: his younger son Henry (OTL Henry VIII) continues his career in the church.

Arthur dies - but the state of his relations with his wife, Catherine of Aragon, is clearer (thus affecting Henry VIII's claims in his divorce petition).

Both Arthur and the younger Henry die before their father. Possibly a 'joint monarchy' involving Margaret and James (as with the Spanish situation) follows.

Henry VIII
His first wife Catherine bears at least one son who survives, so there is no need to break with Rome.

The Pope is more willing to cooperate in granting Henry a divorce, so Henry remains more or less faithful to Rome.

Henry adopts a different attitude towards Rome.

Edward VI
(Edward in OTL inherited the crown as a sickly child, and died young.)

Edward survives longer - and marries Lady Jane Grey/someone else.

Mary I
Mary marries someone other than Philip of Spain.

She is more tolerant towards the new social and political order in England.

Mary has children by Philip (or whichever partner she marries): the Catholic succession continues.

Elizabeth
Elizabeth marries (and has children).

The Spanish Armada succeeds.

An arrangement is made between Elizabeth and Mary Queen of Scots.