LHR is not THAT bad. I do it all the time and although it has gotten noticeably worse in the last few years (especially now with the one-bag rule), it is certainly not the life-threatening experience that some would have you believe.

With that in mind, I will give the following pieces of advice on transiting LHR:

(1) Give yourself no less than three hours to connect--even for a T4-T4 transfer. You will have to go through security, the lines for which can be positively shocking, even for an airside transfer. If you are traveling in J, it is faster since you can skip much of the line, but if you are in C you are really going to wait a long time. At LHR, you will also run the risk of delays in reaching the terminal in the first place--especially in summer, there is a high probability that when you land, there will not be a gate available for your aircraft. Be prepared to sit on the plane for up to 45 minutes after you touch down.

If the TA is trying to give you a shorter connection time than three hours, resist firmly. BA may have frequent flights to GVA, so if you have the luck to be able to get through security faster, you can jump on an earlier one if there's a seat. But don't book yourself onto it ahead of time.

(2) The same liquids rules apply as in the US for carry-ons, and the UK has the added thrill of the one bag limit. If you can possibly manage to avoid carrying lots of toiletries (not too hard, if you are going to stay in a good hotel), and if you can possibly manage to get all your stuff in one regulation-size carry-on (do-able if you have a short trip or are a very good packer), it will be worth your while. If you have to check bags, try to keep it to one bag--less for them to lose, as LHR can be a baggage black hole. (BA is notorious for losing things and then having the world's worst tracking system after they do so.) However, they did surprise me two weeks ago by managing to transfer my suitcase all the way from DUB-LHR-PHL with no problem.