Obtaining an 'S1' from the UK

I have had a house in France for many years. I’ve gradually increased the amount of time I’m here so that now I’m here most of the time. I am enquiring about getting an S1 and have spoken to the UK DWP Overseas Team who have told me what I need to tell them. This includes the date at which I became resident abroad, which is rather moot. What I want to know is what I may lose on the UK side when I register in France. At present I pay income tax in UK as my income is in UK - will that change? What will be my rights to healthcare in UK once I am registered in France? etc. Will my official moving date affect my ability to apply for a Carte de Sejour in France? It is all rather confusing.