For Britain, maybe have an earlier union of the crowns of Scotland and England, perhaps at the time of the Spanish union. This weakens parliament in either state at the expense of the crown.
Then maybe have a more 'bottom-up' reformation in Britain. This makes parliament unworkable in the sixteenth century, as the catholics and protestants won't sit in the same chamber without violence breaking out.
Meanwhile in France, there is a top-down reformation, and the king becomes head of a united church. There will be a long set of wars against the Hapsburgs, but that happened anyway.
Then a French dynastic struggle weakens the crown. A new King (maybe a kind of Henri IV) brings in a parliament, binding his successors. The more absolutist provoke rebellions, so they get shunted aside. Add in an invasion or two, so the parliament-raised army is seen to do better than royalist incompetent ones.
Oh, and add in a British continental connection, like Hanover. This ensures the Brits get involved in lots of wars, which drain the treasury. The more absolute monarchy taxes merchants much more heavily than France, and periodically repudiated its debts.
Regards
R