Luggage storage - Hampton Court

[JamMakingWannaBe]

I have tickets to Hampton Court Flower Festival on 3 July. I'm tagging the day in the middle of a trip visiting relatives so will have two suitcases (1 large for me: 1 cabin size for DD).
Can anyone recommend anywhere near Hampton Court that offers luggage storage? I've tried Stashr and Bounce but the lockers are either too far away or not guaranteed on the day.
Alternatively, I can stop off enroute at either Twickenham or Staines.
TIA.

Will you be transiting through central London? I would leave them at Waterloo or whichever mainline station you’re coming into.

I will be travelling in from my brother's near Reading to my sister's near Kentish Town.