If you're really keen to use the spa, arrive in good time. I don't know the current typical wait time at T5, and it will depend on the time of day anyway, but you might need to arrive 2.5 or 3 hours before your flight. The Concorde Room which all First Class passengers can use is jolly nice anyway and if you've not been there before then I don't think you'd mind spending a couple of hours there at all - many would recommend you do exactly that, spa or no spa.

If you're arriving on the overnight flight then you'll be able to use the Arrivals Lounge and then you can just wait as long as it takes for a spa appointment - which in my experience has usually been around an hour, and typically a bit less than for departures. The 1150 arrival (BA104) might also get you to the Arrivals Lounge in time for a spa treatment, but it's not a sure thing.