I am not in practice so have no vested. interest. However having had to deal with inter-EU employment problems in a multi-national company I would say that this is one area (of many!) for which you really need to take, and pay for, professional advice. In the said company ALL the head office staff were chartered accountants but we STILL took advice on these matters. Asking for it for free is not going to get you the security that you need going forward.
Well, how complicated do you want to get?
Given you are going to be resident in Italy and I presume the work you will be doing for UK clients will be done in Italy (you do not come here to do the work/will have no permanent establishment here) my first thought would be to find out, if say a sole trader registered in Italy, how that would be treated, I might also see how other entities, like Italian registered companies, are treated there.
Accordingly I would find someone (Likely in Italy) who can advise re Italian tax and business. Ideally they might also understand UK tax. I would also, when looking at my options, consider compliance costs, hopefully they can also advise.
My only knowledge of Italian business practices dates from the early 1990s when I worked for a UK company importing clothes from Italy which we retailed here.
Given our Italian contacts (our MD was Italian) I discovered that back then nearly all Italian retailers ran two sets of books and staff got paid two wages-the official wage from the official books and the cash in envelope wage from the "other" books.
Not much use to you but a good lesson in acquiring local knowledge.
You should have a good read of the UK/Italy DTT but assuming it is fairly standard, then once you've left the UK, you'll need to register with Italian tax authorities and you will start paying tax there. Inform HMRC by giving them the cessation date of your trade (i.e. the date you leave the UK). You'll need to complete a tax return for that year and depending on dates you'll probably need to claim relief for the foreign tax. After that you can ask HMRC to stop requesting UK tax returns so you can just concentrate on paying tax in Italy. I think the VAT threshold in Italy is lower than the UK so if you are not currently VAT registered you may well need to think about that too.