Since 2009, France and the UK have a double taxation treaty (https://www.gov.uk/government/publications/france-tax-treaties)
So, you have to pay your yearly tax in the country you spend at least 183 days (half the year+1). Then, you can deduct what you've paid in the other country with the double taxation treaty.
I am familiar with this situation concerning Spain (residence and country of nationality) and the UK and it works fine. However, my yearly income tax is revised thoroughly to see if I made the tiniest mistake. I guess this is the "French authorities may not be happy about it" you read about.
France has similar double taxation treaties with Switzerland and Luxemburg. The last one will remain a member of the EU, but I don't think it changes so much because I had to ask for the refund of taxes here pre-Brexit and I had to prove everything and follow all the steps very carefully.