8.45 Gatwick to 10am haircut:
15 min off plane, 1 hour travel time to bag drop (if you have your ticket in advance, 10 min wait, 40 min train, 10 min to bag drop), 15 min to Trinity square. You're 15 min late in the best case scenario without an airport or train delay.
10-12 haircut and on to Southwark Bridge:
Most stylists expect 2 hours; you are 15 min late, 5 min checking in, need to leave 25 min before 12 to travel, sort make-up (water gets on it during hair appt), check in before your meeting. You have 1 hour 15 in absolute best case. It's your money to waste, but I wouldn't bother adding the stress, just go to London the day before and fit it in then.
12-2 lunch meeting and on to Shard for 2.30:
It's unusual for a 12-2 lunch meeting to finish rigorously on time, most people would allow a lag of at least 20 min. You'll also need a wee at some point!! You'll have to get a taxi to the shard, but it's uncomfortably tight if both clients are important enough for timeliness to matter.
2.30 to just before 5 meeting and back to Gatwick 6.45:
You're allowing 1h 50 for a 1 hour 5 min journey without delays (wait for lifts 5, to bag drop 5, wait for train 10, train 40, walk to check-in 5). Depending on airline you have most likely already missed the check-in. BA won't let you through at under 60 min. London city would let you in at even 20 min before the flight, but not Gatwick.
Ditch the haircut, don't bother with bag drop just carry less, change flight or meeting times a bit, then you'll be fine. Or get hotels each end and change the hairdresser time.