We generally don't do regents in this country. There was the Prince Regent, later George IV, of course but he's not regarded as a wonderful precedent. Plus we already take steps to avoid this sort of thing - it's why the Queen and Prince Charles never travel in the same plane, Charles and William don't either, and there will come a point when William won't travel with George (though that point is a long way off). If there's an accident, we try to make sure that not all the heirs are wiped out at once.
There's really no point in a regency - as GG says, our monarchs don't actually rule so the government would go on doing that and the only technical difficulty would be who signs Acts of Parliament. I imagine there would be a committee of the Privy Council to advise on forthcoming legislation and someone, maybe called 'regent' would do the signing but the title would be more honorary than anything.
The Hubble Telescope has just picked up a sound from a fraction of a second before the Big Bang. The sound was "Uh oh".