becoming an italian consultant (independent contractor) i.e. invoicing the UK company monthly in order to pay taxes in Italy only.
keeping the contract in UK as employee and pay double taxes. If so how to proceed with that considering the Brexit? who to refer to to apply for the double taxation?
If you want to pay low amounts of tax, you are probably in the wrong country.
Realistically, especially post-brexit, you probably need to invoice through a company. There is a corporate tax rate of 31.4%, and a dividend tax rate of 26%. That means for every 100 euros you bill, you will receive 50.76 of them, assuming you don't write off any expenses.
You can remain an employee of the company, but I am almost certain they would need to register to pay tax in Italy, and they would have to pay things like payroll tax. Dependent on how Brexit happens that may or may not continue to be possible. You touched on paying UK tax, and while if you company was requiring you to work abroad you could have some number of years continuing to pay UK social contributions, and UK tax, this would expire and isn't really relevant to your situation.