Not quite sure I understand the balance…you say “working remotely from France” which could mean you are working from home in France, in which case you would be french resident etc etc. But then say “working a compressed week in London” which is very different as you would most likely be classed as UK resident as that would add up to over 183 days a year.
Access to healthcare is individual now, with PUMA. So your wife can get signed on after three months residence as long as she can show adequate resources, and she will have to pay the required cotisations. If you remain a U.K. resident then you can’t get access, and just have to rely on using your EHIC (long may it last).
But it would be worth getting professional advice on your residence and tax status as it can be complex and hinge on odd things. There is an online tool on HMRC website to have a simple check, but you might find it doesn’t come out with a clear answer.
We did similar for a year or so as I stayed in UK to work and look after my mother, and OH became resident here. The french tax authorities struggled with us as we didn’t fit their boxes. And I wasn’t eligible to join french health system.