I may not have been very clear, but you are right, the lift brings you out infront of the BA first class area, and beside the entrance to the Hilton link bridge. However, you do still have a short (ish) walk to the hotel.

One problem I found with using the express between 1 and 4, was that the entrance to the platforms were near one end of each platform, but at opposite ends between T1 and T4. i.e. you got onto one end of the train at say T1 because the entrance was at that end of the platform, but when you got off at T4 the exit was at the other end and hence had to walk the length of the plaform first.