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In the 1870s, at the same time that the Houses of Parliament were rebuilt, Victoria Park Gardens was created. The 1895 map shows that the garden used to be much smaller, only extending as far south as what is now Great Peter Street. The gardens, funded by W. H. Smith, opened in 1881. From there down to Lambeth Bridge, was all wharves, cement works, etc.

Some of this garden was lost when Millbank was widened, but the garden was also extended south as far as Lambeth Bridge and was also widened as part of the 1898 Victoria Embankment extension. The new land was laid out as a garden in 1913-14 and opened to the public on 30 June 1914. Then in 1923 a section of this garden was given over to create the Spicer sandpit. Only 5 years later the 1928 flood arrived.

This Royal Parks report contains photos of children playing in the sandpit in 1927/8 and says "The Playground’s first incarnation took the form of a large sandpit that proved to be very popular with local residents. In later years this was supplemented with, and eventually replaced {by} play equipment more traditionally associated with a municipal play area, namely swings and a small slide."

Google Street View goes back to 2008 where you can see that this screen wall with the sculptures used to be at the southern end of the garden, with the playground between it and the garden, whereas now the wall separates the playground from the rest of the Gardens. 

The 2014 renovation moved the screen wall and remade the playground, once again including a sandpit.

Westminster's 2019 Victoria Tower Gardens Conservation and Significance Statement is very informative with some historic photos, drawings, etc.