However, ridding the UK of monarchy is probably the last step of secularisation. Right now, trying to get rid of the Queen is from a practical standpoint impossible - it's a constitutional quagmire of problems I'm lead to believe.
Actually, there was a group of people who got rid of the English monarchy and established a decent constitutional democracy back in the 1770's. Can't think of the country of the top of my head, but a google search should find it for you.
Joking aside, as long as the monarchy allows the Parliament to run the country unimpeded there is very little reason to dissolve the monarchy (at least in this american's view). While the monarchy is quaint it still ties modern culture to past traditions. As long as the monarchy understands that its role is to serve its subjects I think the institution will continue to be a healthy one.