Are you flying BA?

As long as you get to the queue BEFORE the Transaero from Phuket / Bangkok and the Thai Airways from Bangkok ( both planes arrive around 1600-1800 ) you will be ok.

The morning BA 747 flight from LHR arrives smack in the middle of the arrival of these flights. Expect really long queues, as the Russians arriving back from holiday ( very often in a state of post vacation depression) get through the queues, with grandma, little ones and the wife. All pissed off after being stuck in Economy for so long, often with no cigarettes or booze ( Aeroflot stopped serving liquor on Thailand flights, Im not sure about UN or TG)

At this time, both the downstairs and upstairs lines are used. And they can be long. I would be more comfortable with 4 hours, but its doable with 2 hours 50 minutes.

Check DME arrivals, if the Thailand flights arrive when you land, RUN to the front of the immigration line. Transaero flies a 747 from Bangkok / Phuket and Thai a 777-300ER, packed with passengers.

The best time to arrive at DME is in the morning, before 2PM. At 2PM Singapore from Houston arrives, along with Qatar and Etihad. The immigration line gets progressively worse until about 2000. The US flights used to arrive around noon, which was great!