Ooh go to the South Bank. Me and the kids always go there on a trip to London! Lots of things happening, buskers, street theatre type stuff. People sprayed gold and being statues. I love it! Plenty of places to watch and drink, and have a look at what is going on in the Sotuh Bank Centre, which is super kiddie friendly.
London is full of brilliant things! Check out the grant museum of zoology, its free to get in and there are some great things to look at!
I've done London with 3 kids including a pushchair (one year I did it with baby in sling, toddler in push chair, and walking 5yr old! I'm not going to pretend it was stress free but it was worth it!), but the pushchair might be a hinderance for some things. I'd suggest M&M World, for example, but not sure how easy it would be with a pushchair!
I never had too much trouble with pushchair on the tube, though id advise taking a super lightweight one so if it is busy you can fold it up and hold it if there isn't space.
London is brilliant. We will do our annual trip there this summer too. We stay outside of London though and just travel in each day.