Take a ruck sack with you, that's what I do, they give you a number to check in your bag to a storage area when you pay to get in. You'll be fine in trainers. I'd recommend a wrist wallet if you can get one in time, but if not, they will give you a tiny little plastic bag to keep your money together, most put it in a sock.
At 10:30 it will still be light, it really is right next to the tube so you'll be fine.
I wear my compression gear, singlet or cycle gear under tracksuit bottoms and a hoody when I go so I don't really have to change, just remove a layer :)