My understanding was they still do: I read somewhere (can't find the link now, sorry) that when the system was explained to Tony Blair on his second day in office, he went white.
Firing the nukes after the destruction of the government is 50% of the purpose of Trident. It's a second-strike deterrence weapon. It's not as if it could be practically used as a first-strike weapon independently of the US, there aren't enough warheads to obliterate Russia or China on our own.
http://en.wikipedia.org/wiki/Letters_of_last_resort
It's terrifying that nuclear war was considered likely enough to do this.