Sorry, folks, I know this has been addressed before, but the Search function is giving me only information that is completely irrelevant (other airlines, other airports!) or so old as to be of dubious reliability.

I'm arriving LHR at 0555 on Saturday. Only 1P and upgraded to C, so no arrivals lounge access. The LHR web site is short of details about what's available in the terminals, and practically all Internet lookups say nothing more specific that "showers are available." If I were staying in London, it wouldn't be a big deal, but I'm heading on up to Edinburgh (via train, so departure-side facilities are no good, either) and will really, really need something before that.

On one of my previous trips my seatmate headed over to the Hilton to have a shower. Of course, the Hilton web site says nothing about it, either, so I don't know if that's available only to super-diamond-platinum folks (he was one of those with about 200 nights/year), or if it was in their fitness centre (which doesn't open for a while on Saturdays), or what. I have a first class Railpass, but somehow I wouldn't expect to find showers in the first class lounge at Kings Cross.

Any knowledge?