There is a programme of education about the FIS in progress at present. The intention is to publish an article in the popular aviation press giving a plain language idea of how the FIS works. However, as is often the case with large companies, the progress of the article, which was written a few months back, has stalled whilst awaiting approval for publication from the corporate communications department. It is also waiting for the remodelling of the FIS suite to be complete so that photos can be included to show the working environment.
Pending the article, the rough sketch given by CM is about right.
I would only add that if you are looking for traffic information in some of the busier areas of the UK, then local airfields (even A/G only) will often have more meaningful information than will London FIS. This is because only a very small percentage of those people who are out there actually call London FIS.