BA used to offer this on a personal basis, at a very good rate in the middle of the night. There was no BA training overnight, but the centre was still running as it was a 24 hour place also doing third party work. For that rate, you didn't get a pilot instructor, only a sim engineer - but they were more fun, anyway, allowing you to throw the sim around like it was a light aircraft.

Now, BA only do this as part of a corporate entertainment package.

However, in the UK there are still a number of commercial providers who will resell full flight sim time to individuals. The last one I did was hosted by Alteon at LTN.