Book a visit to the sky garden now. It's free but you have to book in advance.
Do that as your 'central' thing for your friend-free day.
Arrange other things around that.
As previous poster said, do less; slow down; allow time.
You'd never dream of trying to cram a million things into a 'nice' day at home - so why do it to yourselves on a visit to another city?
After (or before) the sky garden, do something fun and low-stress: a walk around Soho/Covent Garden - with lots of stops for food and coffee.
And you could book a musical in the evening.
Remember: it's meant to be fun.
I think, in all honesty, buses are tricky (unless it's a straightforward trip past the HoP or down Oxford Street). They are confusing in a way the tube isn't.
The tube really is very straightforward. As a previous poster said, you can just let a busy tube sail past and catch the one behind.
If you are really panicky, have an 'if we get separated' protocol: getting off at the next station and standing by the wall to wait for other family members to come and fetch you.
But you won't get separated. Your two are pretty grown up and it will be fine.
There's a shop selling just pasta de nata in Soho, just around the corner from a shop selling amazing sushi rolls. It's just behind Foyle's and over the road from a tiny, hidden pocket park.
Load google maps onto your phone and have fun. ❤️