The Dome of Discovery occupied the land where the gardens now sit. The Dome lasted from May 1951 as part of the Festival of Britain, until September 1951 when it was torn down and sold as scrap metal. I then can't find what (if anything) was done with the land until 1977 when the Jubilee Gardens were built for the Silver Jubilee.
I apologise for this very specific question.
My interest in the Jubillee Gardens is at best tangential (there's a fun war memorial there!), but as far as I can tell, the Dome of Discovery was replaced by... a car park. This is based on Ordnance Survey maps from the 1950s and 1960s.
The Dome actually seems to appear on the OS map published in the late 1950s/early 1960s, as the survey of that part of London was done in 1950-1 as far as I can tell. The Dome isn't labelled, but there is a very obvious large circular structure there, so I think we can allow that particular leap of faith. However, in the next survey (for which we have the less usefully precise date range of '1950-62', but presumably some time after the previous survey), the same space is clearly marked as a boring, vaguely rectangular 'car park'.
Sadly, I don't have access to the next OS map in the series, so it is possible that the land found a higher calling between c.1960 and 1977, though to my mind this is unlikely - had anything more impressive than a car park been erected, it would have been difficult to replace with a park come 1977. If you want to go digging yourself, the park lies on the boundary between maps TQ38SW and TQ37NW.
Edit: I suppose I could have saved some time by consulting the obscure source known as 'the Jubilee Gardens Trust website', which confirms the above, but that's hardly the AskHistorians spirit now is it?