The freedom of movement between EU member states is only on the grounds that you make no claim on social security in your first 5 years there. If you do, legally you are considered an economic migrant, and you will be returned home, pdq. Sarkozy did this with Romanian gypsies who were begging and they were repatriated.
The cut off date for UK citizens entering France and being able to affiliate to the 'cheap' CMU system to get a Carte Vitale was September 2007 - brought in by Sarkozy due to frauds by, among others, Belgians and Irish living in the SW having the capital to buy a new luxury car each year, but claiming they had no income....someone ratted on them. This cheap system for someone not getting healthcare through their job costs 10% of income.
It covers 70% of medical care - for the rest you either take out top insurance, or pay as and when. Dd had 10 days in hospital in 2009 - it cost ?8,000 included nursing care, 1 blood test, 2 appts with the psychologist. Her medecines she took in from home. Had we not spent ?2,000 a year on a top up policy, we'd have had a bill of ?1,600.
The alternative is a 100% private policy. They are illegal for French citizens, so there isn't a market, so the few that exist are very expensive.
The E111 only covers you for a 3-month stay, and is for emergency care only. The extra costs, even for an emergency, can amount to ?50/day.
In addition, if you leave the UK for France, intending to be there more than 3 months, your E111 doesn't count.
You may have some limited access to the French system ie get a Carte Vitale if you have an NI contribution record in the UK, but you will still have the top-up policy to fund.
For most medical costs, you have to pay out up-front and then reclaim the amounts back. Hospital accounts depts, the local Caisse Primaire de Maladie, your insurance top-up company can all be very slow to provide the paperwork. I had my gall-bladder removed in mid-November - had ?60 per appt for the surgeon, anaesthetist, lung specialist and cardiologist before the operation, a surgeon's appt after the operation, a ?150 top up fee to the surgeon and ?140-worth of operation compression stockings (that they wouldn't let me wear) to fund up-front.
Say in total ?600 worth - and it took to mid-Jan to get it all back. Our income isn't high enough to pay tax....
There are sites that can give you more detail on the system in France - it's complicated, and can be really expensive if you don't have the right top-up policy.