About leggings...I just made a post to your other thread.
All the hotels we stayed in furnished washcloths and bath towels.
About bathrooms. The pay-type are mainly found in large train stations.
There is a turnstyle, you put coins in a slot, turnstyle turns and lets you go through to the rest room door.
The only place I have used this was in Victoria Station.
Make a point to always use the rest room when you are in a museum, attraction (such as Tower of London), church, pub or restaurant. These are free.
Dressing for the tourist sights. No one cares what you wear as long as you don't look sloppy, i.e., no old T-shirts with ratty shorts.
Yoga pants or leggings with a nice (tunic) top--yes.
There are some photo examples of this in my posting I did on your other thread.
I have seen people in London wearing all types of clothing.
Most tourists are dressed very casually.
At the Tower of London, I saw tourists wearing everything; shorts, jeans, sweat pants, yoga pants.
It's a different matter if you are going out to a nice restaurant in the evening, the theatre, or to a posh department store like Harrod's.
Dressy casual as you would at home for an office party or function.
Black or khaki slacks with a decent top. Or nice jeans with a decent top.
Proper dress is required for afternoon tea, if you are doing that.