I became tax resident in France in 2017, notified the Nationwide of my change of address and that’s been fine. The only disadvantage is that their Flex Account free travel insurance isn’t valid if you’re not a UK resident. However, the only time we tried to claim was when still resident in the UK and my wife went to Jo’burg because her father was ill, unfortunately he died, so she re-scheduled her return flight in order to attend the funeral. Nationwide’s insurance refused to reimburse the cost of rescheduling on the basis that she didn’t have to attend the funeral!
Nevertheless’ one reason that I remain a customer is that inline shopping is easier than with our Crédit Agricole Visa card (which has a cheeky annual fee!) is that CA Visa purchases use an SMS confirmation and we live in at the bottom of a gorge with poor mobile reception.
If your UK banking’s free and there’s only a tiny charge on overseas purchases and withdrawals, it may well be worth keeping the account open. However, you’ll probably still need a French account for local bills and stuff.