London Transport Museum + HAVEN in heart of Covent Garden
Price: £16 for adult, free for children. £16 is for an entire year though (same price as single admission), you just need to show them ID and sign on the ticket.
Where: Covent Garden Plaza (South East side)
What's on Offer: Lots of trains, buses etc to clamber in. Activity packs available for kids to complete. My daughter (2y6m at the time of visit) loved collecting the stamps at the different stations. There are two main play areas; upstairs one is quite small, better for toddlers and babies. There's a bus and DLR you can go in. Also very nice slimming mirror in the photo for a fun selfy. My daughter loved dressing up in the hi-vis jacket, and they have tools so you can fix the DLR. The downstairs play area is a bit more manic with a number of different transport things they can play in. There area a few London/train play tables, with limited number of trains so perhaps bring your own toy car/train.
There's a cafe upstairs too. The chips were nice! Nothing spectacular about it, but they do have kids food (and nice small cheese sandwiches for little ones).
Highlight: For some unfathomable reason, my daughter loved playing with the mop and bucket the most.
When I went: July 2015
Now for the HAVEN! My Top Mums Tip.
Actually, this might just work during off peak hours. However.... It's in the heart of Covent Garden, overseeing the Plaza, with terrace space, baby changing facilities and lots of space for buggies. Where you ask? The Royal Opera House! Our three little ones had a synchronized nap so I even managed to fit in a glass of wine in the Sun. Amazing. Gutted I don't have a nice photo - we went during the great heat wave of July 15 and it was beeeeaaaauutiful.
Sidenote: Sorry the pics aren't great. Not a big fan of plastering my daughter's face on the website so I didn't have many to choose from :)