We are in the same situation, but a little bit ahead of you in the process, having applied and awaiting a decision. No real idea how long that may take…
You don’t need to have formal residence status to apply, just as well because we now can’t apply for a carte de sejour until June at the earliest according to the latest info. However you do have to prove you’ve been living in France for at least 3 months. Note: this means already here 3 months, so a brand new tenancy agreement isn’t enough.
You apply via your local CPAM office, in our case in Tours. However, they don’t process your application, they pass it on to a specific office (CREIC) in Nimes, although they may also ask for more documentation. The application form is straightforward enough, and in theory tells you everything you need to supply with it, but in our experience they’ll come back to you for yet more information. Some of which is difficult to supply - for example they asked us for proof of entry into the country - which we don’t have as up till now passports aren’t stamped (this may change!). However, we’ve written to explain this and so far it looks like they’ve accepted our explanation. Time will tell.
CREIC specifically deal with applications from non French applicants, and historically mainly non EU. You can write, or ring them (on a chargeable number), but frustratingly not email them.
You may need to pay an 8% cotisation depending on income, we don’t yet know ourselves, although like you we are living at present on my pension alone. Obviously this aspect depends on personal circumstances, but even 8% is a load cheaper than private health care.
According to the French govt website, at the moment an EHIC card is sufficient for the first 12 months, so no health cover is required. This of course will change by December, so private cover may be needed by then. I haven’t had to test this in practice…
Initially, you should get a letter of attestation - the Carte Vitale comes later after some as yet undetermined period of time. The letter is all you need to get you started with doctors etc.
A Mutuelle is optional - however most people will get one as it tops up the state provision, and allows you to claim back the costs not covered by the state. How much it will cost is a bit like how long is a piece of string, because it depends what level of cover you choose.
I’m 58, so I have another 9 years to wait for my UK state pension- I very much doubt if S1 will exist by then so I’m not counting my chickens. Time will tell!
Hope that helps a bit, as I say, we’re part way through applying, so maybe soon I can update with progress!