Theocracy is the most offensive aspect, in my opinion, but monarchy is almost as offensive. Well, gosh maybe monarchy is more offensive. It's hard to decide.
While I agree with you completely as a matter of principle, in practice the powers of the Monarch and Church in England are so tightly constrained by constitution and tradition that they are insignificant against other sources of oligarchy or theocracy. In theory, I'd like to see the Monarch abolished and the Church disestablished, but I'd want to be thoroughly convinced that any alternative wouldn't have unintended consequences.
The first thing that happens when the Queen opens each new session of parliament is that her representative, Black Rod, has the door of the parliament chamber ceremonially slammed in his face. The Queen then reads a speech written by the elected government. She is then obliged to rubber stamp whatever laws the elected government passes. Should she refuse, there would be a huge constitutional crisis that would probably end in the abolition of the monarchy unless her action had such massive public support that the government was embarrassed into backing off (in which case, what's not to like?) If any member of the Royal family makes any comment that could be construed as "political" there is a massive political and media row.
As for "theocracy" someone has already posted that the House of Lords (which is where the Bishops sit) only review and amend legislation, and any amendments be overruled by the elected government (at the cost of a certain amount of publicity). The Church of England (certainly the English branch) is about the most liberal non-fundamentailst bunch of god botherers you can find. There's some questionable, like the requirement for schools to have a "broadly Christian" assembly every week, but thats more honoured in the breach than the observance. Its not the UK that agonizes over teaching of evolution...
Of course, I know that the USA with its democracy and strict separation of church and state has no problems with wealthy, unelected individuals having undue influence, or with local government trying to (say) block the teaching of evolution in school...
I certainly know in which countries I'd be most reluctant to publicly declare myself an atheist or burn a flag (should I feel the urge)...