Last time I went through at Heathrow, I was the first in line, and waited for about 30 seconds. (This was in late July). So I think it has to do with your timing/how many flights have arrived before yours. September/October are normally busy because of students arriving, so I'm not sure when Transpondia said it was taking ~3 hours... But if you're concerned about getting through the airport quickly, my tips include:
- Sitting towards the front of the plane, in an aisle seat. I personally don't remember being on any long-haul flights that use the back exit.
- Remembering to bring a pen so you can fill in your Landing Card on the plane, and then storing it with your passport/any other required documentation.
- Travelling light. (At this point, you'll only have your carry on/personal items to worry about, and as you mentioned you're using movers, can I assume you won't be checking any bags?).
- Getting yourself ready to get off the plane
before the final descent (when you're no longer able to move about the cabin). I like to use the toilet, wash my face, etc.
- Removing anything you have stored in the overhead bin (like your coat, which you can sit on, or your bag to put under your seat).
- Putting on your shoes, coat, etc. as soon as you hit ground, and do any final checks to make sure you have everything.
But again, most of these things only really play any part in you getting off the plane quickly if you're in one of the first few rows.
Once you're in the airport, I like to think of it as a race...
- You shouldn't need to use the restroom, so you're already ahead of quite a few people from your flight.
- Walk quickly and pass all those slowly walking dawdlers.
- In LHR (Terminal 4, I think), I remember there's two ways to get down to the ground floor where IO's are. There's a long ramp, or a set of the steps. The set of steps is much more direct, and I've always seen more people taking the ramp (easier for bags with wheels), though both options take you to the same place. So if your Carry On is light enough, carry it.
- Once you see the lines, simply asking the person at the entrance can save you lots of time. When I first entered as a student, they had a special Tier 4 Visa Holders Line, which obviously meant I wouldn't have to wait behind all the general visitors. There may have been fewer IO's for that line, but fewer questions needed to be ask, so the time spent interviewing each person entering the country is shorter. So, it's worth asking whether they have a separate entrance for visa holders (Assuming you have one).
If you've already done all that, and haven't checked any bags (ie- no standing and waiting for your bag to come 'round on the carousel), you can just head out and meet your eager relative.
Good luck!

(Sorry for the super long post!!!)