The British take their royal traditions with a grain of salt. The royal family and the various ranks of minor royals are lots of fun and part of the history and heritage of the UK. But even dyed-in-the-wool Royalists are not above joking about them or taking a light approach to the whole subject. So don't be shocked by the flippant attitude toward the Queen, her children and grandchildren that you may see on television and in the press.
But do be careful about making jokes yourself. It's a bit like jokes about your family - it's only okay if you do it. So until you know the lay of the land and the feelings of the people you are with, best not initiate royal humor yourself. And, if you are introduced to someone with a title - which you may well be, there are lots of them around - don't fawn or wonder if you should curtsy. Just treat them with the same respect you'd give anyone else.