In a cross-border employment relationship, beyond visa/work permit issues, there are also issues with social security/national insurance and with taxation.
If you live and work in the UK, to be employed legally, you need to:
- Have the right to live in the UK - no problem, you’re a British citizen
- Have the right to work in the UK - ditto
- Have your employer be registered with HMRC
- Have your employer declare your employment to HMRC
- Have your employer file PAYE returns, deduct NI and taxes from your wages, and pay them (and their own share of NI) to HMRC.
Unless your employer already has a subsidiary/branch in the UK, I doubt they would be willing to go through the hassle just for you.
What usually happens in such situations is that you are not employed, but under a freelance contract: you register as a sole trader, file and pay NI and taxes. Instead of receiving a salary, you send invoices to your customer (not employer). Note that depending on the amounts you may also need to register for VAT, collect VAT, file VAT returns, and pay VAT to HMRC. Depending on the situation it could also make sense to register a company rather than as a sole trader.
The UK have however rules (IR35) to prevent people who are in practice employees (they have a single “customer” who is really their employer) to act as sole traders, not sure how that works in cross-border situations.
There are also alternatives with companies that act as intermediaries (so you have a short term contract with that company, and they in turn invoice the “customer”). See “portage salarial” in French or “umbrella company” in the UK.
In any case, someone will have quite a bit of paperwork and admin to do, so it’s probably not worth the effort unless you expect it to last months.