Ultimately I think this means the OTL 19th C. claims about celtic Christianity being separate and the CoE having an independent descent from them get more widespread and get promulgated earlier.
Maybe this ultimately leads to a greater interest in some kind of rapproachment with the Orthodox?
What does this do to England's relationship with Ireland? Does Irish nationalism still take on a heavy religious basis, or is the difference between Catholicism with the Pope and Catholicism without enough to make religion still a convenient dividing line?
I don't know when the POD is, but unless it changes events in Scotland somehow, likely Scotland still goes Presbyterian as OTL. I wonder if you could get a weird result in TTL where England conquers Scotland, England and Ireland end up in a relatively problem-free union, but Scotland is the country where all the troubles and rebellions occur.
If you look at the situation between Catholics and Orthodox, they have a lot of theology in common and a long history of mutual antagonism, so I could see religion still being an issue in Ireland.
As for the Scots, IOTL Elizabeth sent an army north to support the Presbyterians, so I guess the question partly depends on whether a CWOTP Elizabeth would do the same (under the principle "My enemy's enemy is my friend") or stays neutral (under the principle "I hate both Papists and Protestants, and I'm not lifting a finger to help either of them"). If the latter, and if without English support the Presbyterians end up getting defeated, you could conceivably see a situation where a Catholic James VI inherits the throne of England, which would be, erm, interesting. Since I don't think any attempt to rule a Catholic Scotland and Anglican England would be very successful, the most likely scenarios would be:
(1) James adopts Anglicanism to get his rule accepted in England, and forces the Scottish Church to adopt Caesaropapism too.
(2) The English change their succession laws to stop James becoming King in the first place, or overthrow him shortly after he takes the throne.
(3) James somehow manages to bring England back into the Catholic fold, probably via a Gallican-style concordat whereby the Pope's authority over the English Church is officially recognised, but the King holds all the real power in actually running it. How feasible this is probably depends on how much bad blood there is between Anglicans and Catholics by this period (if there's not too much, and if the official re-Catholicisation of England doesn't change much on the ground level, people might just go along with it) and whether or not there are any plausible alternative candidates for King (if there aren't, any attempt to depose James or remove him from the line of succession would run the risk of civil war, which might induce enough people to accept rapprochement with the Pope for the sake of political stability).