Pro tip if you decide for your next flight on UA into LHR to take only carry-on to avoid luggage carousel problems....

Just before the immigration hall (after long hallways, escalators, etc) you arrive in a central space where you can follow signs to the left to Terminal 2 Arrivals or follow signs to the right to Terminal 1 Arrivals. If, despite quizzical looks from staff on hand (you assure them you know where you're going), you march to the right, you arrive immediately at the conclusion of the previous long T1 walkways to immigration, straight into the T1 immigration/customs hall.

In my experience this is usually the fastest of all Heathrow immigration lines. Afterwards it's a quick walk/escalator out the front door of T1. Not glamorous like T2, but much faster to taxi, Heathrow Express, etc.

But don't do this if you're being met at T2 (unless you tell them to meet you at T1 – also easier).

This will work at least until all Star Alliance airlines move from T1 to T2, scheduled for 22 October 2014.