This occurred to me when I realized that James' first wife, Anne Hyde, was the only British royal spouse-parent between Lord Darnley and Elizabeth Bowes-Lyon who was of British birth. And her blood died out of the royal line.
So... WI James' children by her (Mary and Anne) were instead Martin and Andrew?
ISTM they would be raised Protestant.
Their marriages will be different of course. (Later, for one thing.)
But otherwise, the first predictable change may come if/when James' Catholic son is born. There will be no succession issue over it - no tales of a "warming-pan baby", no Monmouth rebellion, no Bloody Assizes. James, though still a fool and a bully, may hang on to power till his death in ~1700.
So - no Glorious Revolution, no civil warfare in Scotland and Ireland. No quasi-union-of-crowns with the Netherlands.
England does not join the Grand Alliance against France.
Martin and/or Andrew probably beget children, continuing the House of Stuart.
So no House of Hanover and no personal union there either.
And the British royal family is a lot more British. (George VI's nearest British ancestor was his 8-greats-grandmother Elizabeth Stuart.)
Or maybe not that much more. George I was Elizabeth Stuart's grandson. But she was half-Danish; so he's ~1/8 British. James I was almost 3/4 British; aside from his maternal grandmother, his only non-British ancestry is his 3-greats-grandmother, Henry V's widow Catherine of Valois, and 3/4 of his 6-greats-grandfather John of Gaunt. But still that brings George I down to ~3/32 British.
Martin II would be Anne Hyde's grandson (1/4), and James I's 2-greats-grandson (~1/32). So Martin II is ~9/32, three times as much.
And of course, unlike the first three Hanovers, these later Stuarts would all be raised in England.
Any other thoughts?