We know that the British prime minister (Is he the prime minister of all of the UK, or just of England, BTW?) chooses the Archbishop of Canterbury, and tells the Queen who formally appoints him.
Now this isn’t a problem for the Queen, since the monarch has to be an Anglican, but what if the prime minister isn’t? (Hasn’t this situation come up - wasn’t Disraeli Jewish?)
I presume they skip lightly over the more de facto part of the situation, but who would actually decide in such a situation?
Disraeli was Anglican. He was born Jewish, but converted when he was a boy, along with his parents.
Anyway, I’d imagine that, even in the cases of Anglican PMs, the Prime Minister sends his reccomendation to the queen only after staffers first give him a reccomendation…that he doesn’t rely on his own theological knowledge.
The PM is given a shortlist of two candidates and must select one of those. As Captain Amazing said, s/he then takes advice over which one to choose ( though it is said that Tony Blair took a rather more personal interest in the matter than has been usual).
The situation has already arisen - Lloyd George is the example that comes immediately to mind - and the answer is that it makes no difference at all. In such cases, the PM can make a recommendation on whatever basis he or she thinks appropriate.
This is, it is true, now subject to the procedure mentioned by Jabba, although, just to be pedantic, the PM is not obliged to accept either of the names, as he/she can return the shortlist and ask that new ones be submitted.
There were no vacancies in either archbishopric while Lloyd George was PM (1916-1922).
Davidson was Archbishop of Canterbury from 1903 to 1928 (click on the “succession” link), while Lang was Archbishop of York from 1908 to 1928, when he was translated to Canterbury.
I suppose Lloyd George may have been involved in the appointment of a bishop or two during his stint in office.
I remember when Jim Hacker was appointing the Bishop of Bury St Edmunds he received a recommendation with 2 names from the Crown Appointments Commission.
Sir Humphrey: “The Church is looking for a candidate to maintain the balance.”
Master of Baillie College: “What balance?”
Sir Humphrey: “Between those that believe in God and those that don’t.”
Point taken, in that the OP did ask specifically about Archbishops of Canterbury, although the constitutional and ecclesiological issues raised by the appointment of other bishops are exactly the same. For the record, during his six years as PM, Lloyd George made nominations to the sees of Bath and Wells, Carlisle, Chester, Chichester, Coventry (twice), Durham, Hereford (twice), Lincoln, Manchester, Oxford, Ripon, St. Albans, St. Edmundsbury and Ipswich, Salisbury, Southwark, Truro and Worcester.
Thatcher was brought up as a Methodist but had become an Anglican by the time she became Prime Minister.
Bonar Law is the other obvious example of a non-Anglican PM. He made one episcopal nomination, appointing Headlam to Gloucester in early 1923.
Phoney Blur is an Anglican. His wife is a very devout catholic and his children have been brought up as catholics.
It is commonly assumed that he will himself convert on leaving office.
His non-conversion before then is because of the politics of Northern Ireland. Or so the rumour goes.
Technically the CoE believe themselves to be catholic rather than a fully protestant church, but that is nit picking theology. The Queen is the head of the CoE, not the Pope.
Lloyd George’s religious beliefs are a bit difficult to pin down because, as a adult, he drifted away from any form of organised religion. He had been brought up as a member of the Disciples of Christ. The one common thread to his religious views was his distrust of the Church of England and he played a leading part in the disestablishment of the (Anglican) Church of Wales.
Andrew Bonar Law was a member of the Church of Scotland. (His father had been a Presbyterian minister in Canada, but Bonar Law himself was brought up by relatives in Scotland.)
I should add that I would be very surprised if Ramsay MacDonald was an Anglican (it would be one of the most interesting things about him if he was) and I have a vague feeling that James Callaghan may not be one either, but, in neither case, can I find any firm information.