Well, with US passports you can go all the way landside and then take the regular shuttle over to T1, but it seems that the on-demand bus is actually reasonably signed now. I'd say it's still likely to be fastest given you don't have to do passport control on either side and transit security at T1 will be much lighter than regular security, plus you don't have to deal with getting around on the M13 to get over there.
These are the signs you're after right before getting to the manned immigration lines (so non-EU line). It takes you down to the immigration posts for people coming off of hard stands where you can pick the bus.
All in all a pain, but seems pretty comparable to an inter-terminal connection at LHR.
