Britain is rather officially Anglican (though with religious tolerance enshrined into law), and i've seen people occasionally create timelines and scenarios whereas Britain stays in the Catholic fold from the 16th century on in, or breaks with "crown religion" altogether under a surviving Puritan Commonwealth or Protectorate in the 17th century.
So, i'd like to shine a spotlight on the Quakers or Friends, a dissenting, sometimes pacifist Christian group formed in Britain in the middle 17th century who particularly thrived in the American Pennsylvania colony until fizzling out numerically by the 18th. Could this sect have "conquered" Britain? Could they have established themselves as the politically dominant group in the country and its empire as a whole? Could they have converted a sympathetic monarchy, or monopolized power in a republican government? PoD anywhere between 1600 and 1800.
 
Difficult because they were quite disestablishmentarian to begin with. That is they were against the idea of state churches, and organised churches in general.
Maybe a more presbyterian outlook during the Wars of the 3 Kingdoms and inclusion by the Protectorate would help. Could bring Scotland more on board and make things less an English Puritan affair.
If we also add in a continuing but tolerising Protectorate we might get this Presbyterian Quakerism dominant in parliament and spread it from there.
 
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