On 23 March the Government published stricter guidance on staying at home and away from others that superseded the advice in place at the time of writing on 19 March.
The good news is that free-to-enter-sites will remain open to visitors.English Heritage say that this is because:
These sites have large open spaces in which visitors can maintain social distancing and they are often located in quieter spots away from crowds.
The closest free site to the Whitley Pump is the Roman town of Calleva Atrebatum at Silchester.
Update at 24 March
At this time Government advice to reduce the transmission of coronavirus is being constantly updated and you should consult the latest guidance on the Government’s coronavirus webpage.