There is no reason to change your website when you move to France. I sell jewellery online and my website is still a .co.uk site that I created before we moved here.
There are logistical hurdles like postage to think about. When we first arrived, my husband was going to continue his online book sales but soon realised it was a no-hoper as the cost of posting books from France to the UK is prohibitive.
If you feel your market will continue to be in the UK (my silver jewellery has little appeal to the French but still sells to a UK audience) then I would recommend keeping your virtual presence there.
Once you live in France, you need to register your activity either as an AutoEntrepreneur or one of the other statutes (and that's a whole other conversation!) so that you are what is known as fiscally resident in France - ie paying your taxe and cotisations here and not in the UK.
With regards to the UK bank account, I too would recommend keeping it. You just need to let them know your change of address. They don't seem to have a problem with existing account holders moving abroad.The difficulty arises when you live abroad and wish to open a new account in the UK. A lot of people have found this to be impossible.
Hope this helps and best of luck! For what its worth, it is well worth making the move to live in France. We love it!