I'm always disappointed when dealing with Heathrow, considering it should be a world class airport their version of the TSA area is just a nightmare. At least in Terminal 3, the room is small and jampacked, the ceiling is low and missing its tiles, and it seems like everyone in line has some sort of problem or another. The workers aren't especially happy or friendly, and I don't blame them.
Ironically the area with ticket counters where you first walk in is quite nice and doesn't prepare you for what is to come. And then after you survive the investigation, you're dumped first into a huge crammed dutyfree area and then a huge crammed seating area. If you can at all wrangle your way into an airport lounge, you really need to. Just be aware that the Amex lounge is apparently a latecomer since it's way, way off by itself (they're improved the signage at least since my last visit). And once you finally get there, it is also small and cramped but not as bad as outside.
They play the same "game" as train stations, gates are not announced until about 45 minutes before a flight. And then you're off to the races because the walk between gates is very, very, very long (and the people movers weren't working). Felt like we had walked halfway home before we got to our plane.
My two taxi rides were each about an hour, so use that as your planning baseline. I had a reservation at the airport Hilton the night before my flight but decided to take a chance that morning taking a taxi from The City and it worked out well (most morning traffic is heading INTO London).