Hi JB,
You will need to review the UK / Italy tax treaty here:
UK/ITALY DOUBLE TAXATION CONVENTION
Article 15 - Dependent personal services advises that salaries that arise in the UK by an Italian resident are only taxable in Italy.
However, it goes on to state that if you are in the UK at the time you undertake that work, then it may be taxable in the UK.
As a UK national, you are entitled to claim personal allowance, no matter where in the world you live, so you may not have to pay UK on the salary. Your employer would still need to set you up on their payroll and report your employment to HMRC.
Thank you.