Tube will get worse throughout the day as the trains warm up. Lines that are mostly outside will be the worst. In general, avoid the Piccadilly, Central and Northern lines because they're super busy and use older trains with less space.
Today I'm going down the Thames on a boat. No idea what to expect, but it's £30 entry so I hope there's food and drinks :\ Hope it's cooled down a fair amount by then, because being stuck on a boat on the Thames in this heat is my worst nightmare. If it's in the sea then that's fine... damn I want to go to the beach now.
Get a Zone 1-6 travel card, they're like £8 or something. Best thing to do in the future is get an Oyster card and top up when you go to London. It'll cap your daily travel the same as a travel card, or less off peak including weekends?
Loading bar isn't that great. It's not a good space at all unfortunately, but it's in a great location. PS if you want a cheap pint, go to the White Horse just down the road back towards Piccadilly Circus and you can get a pint of ale for £2.90 or a lager for... umm... £3.10? I dunno.