Free tourist attractions - museums, art galleries, historic houses etc.
Big parks - we make an all day trip to visit ones further afield that we've never been to before. We do a bit of research and check if they have splash areas etc so we'll bring swimsuits.
Children's Centres - we visit ones in neighbouring boroughs and some of them are open at weekends. My eldest is technically too old but we just fudge her dob on the registration.
Cheap visits to the zoo (for UC recipients).
Walks along the Thames.
Outdoor sculpture walks or trails.
City farms.
Council leisure centre soft plays (not free but 70% cheaper with a concessionary card).
Tourist areas like Trafalgar Sq and Covent Garden - often entertainment and interesting stalls.
Events like Changing the Guard.
Beach trips.
Toy shops, if you can resist buying the big shops like Hamleys and Selfridges have lots on demo.
Local festivals - I check the council websites and community FB for all the nearby boroughs and go to free festivals, they often have food, entertainers and fairground rides.
In the summer holidays all the councils have some events and activities to keep kids occupied, including some free play schemes for those on a low income. I check out all the ones nearby as some areas have much better events and it's easy enough to travel to them.