No you don't have to go through customs. Unfortunately, LHR requires screening for connections so follow the signs to transfer center or whatever they call it. Basically, you will have a long walk and then re-enter the transit area. There will then be a *A or UA desk at the transit check-in area or you can probably proceed to the RCC.

I haven't done this since it was renovated since, I tend to overnight in London when I pass thru. The walk to customs appears to be longer than in the past..... and I think it's the same for connecting pax.

They have finally segregated arriving and departing pax, which was like a 2 year project, but even if you made it in to the RCC they would not allow you to check in.