Our son and his family joined us in London for a week in May, 2016 when our grandchildren were ages 22 months and 4 yo. We had an apartment near Tower Bridgge on the south side of the Thames, and the kids loved being able to run and play in the open area of the Thames pedestrian path (The Queens’s Walk), especially around the London City Hall. There was a water feature with a “river” and spray jute’s that has since been removed, but it is still a nice open area, with Potters Field Park right behind.
The places around London that they especially enjoyed were (1.) The playground in St. James Park, and the lake and waterfowl there; (2). Kew Gardens, where they could run around and play, especially the Treetop Walkway:
https://www.kew.org/kew-gardens/whats-in-the-gardens/treetop-walkway
(3). The wading pool in the courtyard garden at the Victoria and Albert Museum, and lunch at the cafe there. This was closed during the pandemic and remained so for awhile, so I had to search for information on the current status. According to this list of wading and splash pools around London, it is open in 2024 (see #21 on the list). Museum entry is free.
https://kiddoadventures.com/2024/04/25/over-90-of-londons-best-paddling-pools-includes-splash-pads-fountains-lidos-more-2024-guide/
https://www.loc.gov/resource/vrg.12952/
The Princess Diana fountain (#2 on that list) was not open when we were there, so the kids did not experience this one. But it is open now, and highly recommended by others.
The kids (and we and their parents as well) enjoyed boat rides on the Thames and the London buses for getting around; the Tube not so much (too noisy and often difficult to access with the stroller).
We took them to the London Transport museum but the exhibits and activities did not interest them (they are geared toward slightly older children), so that was a waste of time and money for us.
Based on our experience, I would recommend the South Bank/Southwark area, or just across the Millennium Bridgge near St. Paul’s: or the neighborhood just south of Hyde Park/ Kensington Gardens (not sure if that is Kensington or Knightsbridge). They can look for laser ice apartments in these areas, such as Marlin Queen Street (close to St. Paul’s), Native King’s Wardrobe (pricy), The Harrington Residences (South Kensington), or others. It is easy to search for apartments on booking.com to find options, then go to the property’s own website to book.