They're both, as is the US. They aren't pure democracies, though.
Since there is, more or less, no other type of democracy around when a colloquial English speaker says "democracy" they mean representative democracy. Everything else is a pointless annoying semantic game.
Representative democracy is a perfectly acceptable term. Trying to shoehorn monarchies into the term "republic" is absurd.
At this point, the Queen is nothing more than a vestigial organ. (as an example, did the Queen come out for or against the Brexit vote? Have you ever seen a head of government remain so quiet about such an important event?)
I disagree with little of the above, but I wonder how much of it is due to 'Liz 2 in particular. I can imagine a different monarch exerting a great deal more influence. They'd be perfectly able to do so. The UK's armed forces are loyal only to the monarch and not the government, for example.
Well, I would hope the move is towards more democracy, but it seems people everywhere are eager to vote against democratic freedoms these days.