London Heathrow is probably the worst airport I've ever connected through.
1) There is no train/tram to connect between Terminals. Waiting and riding the bus takes too much time. I've been to many other airports with much more efficient cross-terminal travel.
2) Security is horrible -- at least 50-60% of bags that go through the checkpoint have to be manually checked. I'm fine with having higher security but there needs to be more staff manually going through bags. I've had to wait upwards of 20 minutes after I go through security to wait for my bag to be manually checked. (This doesn't include the 10-30 minute wait to even put your bag on the conveyor).
3) Flights don't have a gate number until the gate "opens". You won't see which gate you need to go to on the screens until about an hour before. This means you really have to rush to make it to your gate. The truth though is there actually is a gate way before an hour -- you just have to manually go to customer service and "query" for it.
4) Delays, delays, delays. There are lots of weather related delays - fog, rain, whatever. In addition, it seems that LHR has problems directing air traffic. This means you can almost always expect to circle a few times in the air before landing, or wait for a "slot" to open up so you can depart. This can delay you from 5-10 minutes or even a few hours! I've even been stuck in the plane waiting to depart because "air traffic control computers crashed and so they have to manually route traffic".
With the above said, if you do find yourself at LHR find the "Eat" cafe and grab yourself a sandwich (specifically, the Ham and Jarlsberg) -- it is amazing.
The only upside for me is LHR is the only airport I can fly to non-stop in Europe from Phoenix. But with all the problems I've had here I might be connecting through somewhere else next time.