The UK & Ireland forum has plenty of suggestions on what to do.

At LHR, enter the UK through T3 and use the AA arrivals lounge for a shower and breakfast. You won't be able to use the BA arrivals lounge in T5, and the AA lounge in T3 is quite satisfactory.

Then proceed into the Heathrow Express (if you want to spend money and go to Paddington, an area of London largely devoid of interest) or the Tube (if you want to go to the rest of London). Do what you wish to do, then return to LHR T5 at 15:30 (unless you like rushing) and go through (First Wing) security to the lounge, then onto Cairo.