UK global consultant, living and working in different EU countries

I am due to move to Barcelona after living in Paris for 3 years, as property is cheaper there, but still close to international airport, which is ideal for my work.

I now find myself in a situation that potentially I will be working for a French company on a permanent contract.

My first question is regard to who I pay tax to, given the double tax laws?

Also, given Brexit, I can get a residency permit in Spain which allows me to live and work.

Are rights rights conferred as I have the right to work in one EU country??
Or do I need to apply for one in both France and Spain??