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".