You can’t apply for permanent residency after 1 year. For permanent residence you need to demonstrate 5 years legal residence. Provided you can show you were resident on the cut off date you will be given a temporary titre de séjour to allow you to stay and accumulate the necessary 5 years, after which you apply for the titre de séjour permanent.
What is not clear, is what will be required to prove residency before Brexit. (You keep mentioning “fiscal residency” but AFAIK the rules are not concerned with fiscal residency, they are concerned with residency in the broader sense, ie France is your foyer/the centre of your economic interests/etc. Fiscal residency is in any case not a thing you can “declare”, it’s something that is established as a consequence of meeting certain criteria over a period of time, so it is normally established retrospectively, ie after you have submitted your first tax form.) Up to now, for intial applications préfectures have been wanting to see evidence of “stable and regular” residence including for instance justicatifs to show that applicants are in the French healthcare/social security system, either via an S1 or via paying cotisations, stuff like that. However I just checked the Brexit legislation and I didn’t see any reference there to “stable” so probably you are right and they cannot insist on all these documents. The question is, what dossier will they require from people who arrived immediately before the cut off date, because obviously they will want to be able to distinguish between people who live in the UK and who popped over to France the day before Brexit for a brief stay at their holiday home or in rented accommodation in the hopes of blagging a titre de séjour (because you can bet there will be people who will try their luck), and people who genuinely have moved their country of residence and their move just happened to coincide with Brexit? I am not suggesting that you are trying to blag anything, all I am saying that if I were in your situation I would be looking to cover my back by establishing myself here asap. France has a track record of going through paperwork with a fine toothcome because it is keen not to leave its systems open to abuse, and that means that sometimes genuine applicants have a hard time convincing the authorities that they are genuine, simply because for whatever reason they don’t have paperwork to prove their situation.
Of course it could be that I am being over cautious and there will be very minimal checks for the temporary titre de séjour, but that may depend on how accommodating or otherwise the UK is with its EU residents.