I was in exactly the same situation ealier this month.
...1P in upgraded C arriving at LHR T3... and needing a shower and change of clothes before I went to a meeting.

But they let me buy a day pass for £25 at the counter, and I was then able to take a shower and have some breakfast. Decent set-up. Clean and quiet!

That £25 didn't seem a bad deal at the time, but as the previous poster mentioned, it translated to $52.09 in real money, which now seems a tad excessive.

Ants