The following news item has just been published on the government’s official site (see translation below):

Are you British and live in France? In the context of the United Kingdom’s exit from the European Union (EU), do you have any particular steps to take to continue to live in France after the transition period that ends on 31 December 2020? Service-public.fr will answer you.
British people and members of their family already settled in France or coming to live there before 31 December 2020, who are beneficiaries of the UK withdrawal agreement from the EU, will be able to continue to reside in France without a residence permit and to work there without a work permit until 30 September 2021.
Possession of a residence permit will be compulsory from 1 October 2021.
It is necessary to apply for a specific residence permit “UK withdrawal agreement” before 1 July 2021.
For this purpose, a tele-procedure for applying for a residence permit for British nationals residing in France before 31 December 2020 has just been opened by the services of the Ministry of the Interior.
To apply, you will need to send a photographed or scanned copy of your passport and one or more other additional documents depending on your situation.
After validating your application online, a confirmation will be sent to you by email with a registration acknowledgement number. Once your file has been processed, an e-mail will be sent to you by the Prefecture’s services in order to fix an appointment to finalise your application (receipt of the photograph and fingerprints).
Your residence permit will then be sent to your home address by post.