The UK is not a proper democracy. The unelected House of Lords and the unelected monarch being the first giveaways of that.
I'm going to have to call you out on the House of Lords point.
Having an unelected second chamber that is able to be directly overruled by the elected first chamber in extreme cases is the best system. Contrary to popular belief, the HoL is mostly filled with experts in various fields; for example, 1/3 of the Lords have science degrees, compared to just a single MP. I, along with a lot of other people, appreciate having more than just the views of career politicians represented in government.
If we elected the second chamber as well, we'd just end up in constant gridlock (just look at the United States), as members of both houses would refuse to compromise due to wanting to get re-elected. The only reason this doesn't happen all the time in a lot of European countries is because most of them don't have First Past the Post like us, and so don't have 2 parties dominating the show. The US do, which is why their politics is a mess. Even with Proportional Representation, everything works at a snail's pace.
There's a pretty damn good reason for keeping the queen, too. She keeps democracy in this country relatively stable. (Hear me out on this one.) In countries where there are constitutions designed to keep democracy in check, there is a pretty-much universal trend of that constitution being misinterpreted, abused, changed and otherwise maltreated by Machiavellian politicians. The fact that the British "constitution" is based on precedent over hundreds of years rather than a physical document makes it actually much harder to do anything against it. There's almost always a backlash by the Commons and/or the Lords on the basis of this precedent, and it's very difficult for any single politician to abuse the system. On the other hand, with a constitution, if a court rules in your favour that your crazy interpretation is technically valid, you're free to do what you like; this is not uncommon in many countries around the world, especially when the courts are in the pockets of politicians.
With regards to the queen herself, though, she's the perfect way to put dangerous power in a politically castrated body. If she's the only one that can strike down a law for any reason, and there'd be outrage if she did so, then there's no way any Prime Minister can gain an absolute veto as people would see it as an infringement on the sovereignty of the queen's power. In other words, that power is effectively unusable by anyone, and for good reason. This is only one example of several that prevent politicians in this country getting too big for their boots, unlike several republics I can think of. Another one would be the declaration of war without mandate. I suppose one argument would be that there would be outrage if a politician did it as well, but the difference is that a) the queen can't keep her use of it secret, and b) a politician would make sure to have a physical constitution on his side before doing so, which would be inevitable if we became a republic. The public's outrage would be impotent.
If you think politics is bad in this country, just look over at our friends across the pond. The ones that certain special-interest groups here seem so keen to emulate.