Where can I go to sit in the cockpit of a Boeing?
There is a company called Flight Experience which builds 737 simulators and offers their use to the public. These are fully certified simulators and are frequently used by commercial pilots for currency training.

Their closest location to London appears to be in Paris.

I had an opportunity to fly one of their simulators a few years ago, and it was a lot of fun. We spent some time doing landings at the old Kai Tak airport in Hong Kong.