There's more than one of them - there's a constant shuttle between Guys and St Thomas's, and then there's a round trip shuttle between various offices (including those opposite Ambulance HQ on Waterloo Rd). So you're all correct....
It was great when my daughter was being treated at Guys though we live near Thomas'. Plus I felt a bit naughty sneaking on a staff bus.
There was another bus that went from near Lambeth Road to the Chelsea and Westminster. I knew people who used to sneak on to go shopping. Since the 360 route was created we can get to Sloane Sq for our shopping, so have no idea if it still runs.
Nonetheless, it was (and indeed is) a great bus to catch up to Whitehall - you could always get a seat, even if the route was/is somewhat roundabout to take in the various DH (the DOH has been dropped for obvious reasons!) buildings.