When ol' Lizzie died, the Archbishop of Canterbury let us all know that God dropped him a message to say the King Charles should succeed to the throne, which worked out awfully lucky with what was already being planned. The Head of State of the United Kingdom is claiming to derive their authority from God (aka Divine Right of Kings), state and church are officially unified and clerics are a required part of the legislature, does that not make it a Theocracy by any reasonable definition?
To say they 'stay out of politics' is a hell of an overstatement. The Royal Family in the UK have many, many times refused to pass legal bills until they were amended to suit them. Mainly for the purposes of excluding them from any taxes, responsibilities or regulations, which is why having the royal estate as your landlord is actually awful - they're not really beholden to anything. And that's only the most direct intervention - they're regularly consulted by the government for direction and advice, and of course have many other mechanisms of power that are still very real.