Just reporting back on my experience two weeks ago...

We arrived on BA247 (which lands at about 5:20am). There was a very short line at immigration, although we did walk fast to beat the majority of the passengers to the queue. Our bags arrived rather quickly and we weren't stopped at customs. I would echo qfrodo's suggestion to miss out on the duty-free shops. The signage implies that the exit is for "Crew" if you walk straight, and for passengers if you turn right, but don't be tempted to walk through the duty-free shop because it takes a long time to weave through all the passengers.
In the end, we had about 2.5 out of our 3.5 hours to spare, and we walked between T3 and T2 instead of waiting for the shuttle.

I know this is just one experience and isn't necessarily indicative.