The Brexit transition period has now ended, and people who have been legally resident in France up to the 31st December 2020 have until 30th June 2021 to apply for a Carte de Séjour.
Applications are made via the government official website, the process is relatively easy, depending on your situation, documentation is minimal and a big plus for many people is that there is an excellent English version.
Apply here:
https://contacts-demarches.interieur.gouv.fr/brexit/brexit-demande-titre-sejour/
For comprehensive information you should visit the website for Remain in France Together.
https://www.remaininfrance.fr/
If you are intending to take up permanent residency in France, from the 1st January 2021 you will need a long stay visa. You will need to apply for at the French consulate. For more information, visit their website.
https://uk.ambafrance.org/-Consulate-in-London-
If you wish contact them by telephone here is the number: +44 (0) 2030 400 460