Apparently you can. I googled it!

I think the ban is against willy nilly use of hosepipes watering the garden. Don't think it can be "illegal" to fill up water butts and the like using a hosepipe from your tap.

I really don't know about using well water in the UK in times of drought - it's perfectly permissible here, but you will find that using a hosepipe and a pump will empty your well dismayingly quickly, and if the water table is low, it could take some time for it to fill again.

As I understand it, when hosepipes are banned it is banned to even use a pump and a hosepipe to empty your used bathwater. you can leave a tap on and waste lots of water indoors but using two feet of hose to fill a watering can from a tap can theoreticaly be illegal. If you have a legitimate well (you do need permission to sink boreholes or wells) then I believe that you still need to avoid hosepipes or sprinkler systems.
I don`t think it is illegal to pee on your garden yet (as long as you`re not overlooked), but perhaps the waterboard own that too!

You can use a hosepipe from your own private well. The hosepipe ban is surely to stop you using the mains water - not a pump and hose from the bath. How can that be illegal.