You will need to approach the GP first, and take it from there.
In theory, this is true even if you wanted to go privately (self-funded outside the NHS), as the GP would still need to write you the referral or introduction letter. Most private consultants in the UK do work for the NHS as well, so it is likely the GP would know of them, and something about their reputation, even if it is not your choice to see them on the NHS.
I believe all EU citizens are automatically covered by the NHS, and in theory the cost is recovered from the home country (but not from the patients as individuals, so you should not face an unexpected bill back home),
If you can afford to go privately, and would like to, the chief advantage is speed. As you are very likely to see the exact same consultant in a private clinic that you would have done on the NHS anyway, there is not likely to be any REAL difference in clinical approach or standard of care. Typically, a private clinic has nicer, more modern surroundings and nicer coffee, and you get to spend a bit longer with the consultant, but there won’t be any major difference in how they treat MS.
Most people in the UK who are diagnosed privately (I’m one of them) move back to the NHS for ongoing care and management anyway - partly because the expense of staying private after diagnosis is usually NOT met by insurance companies, and can be prohibitive, but partly because there is no real advantage in staying private after you know what the problem is. There is not likely to be any knowledge or treatment available privately that is not available on the NHS - because it is mostly the same neurologists!
One thing I am not certain about is whether the NHS can accept a diagnosis from an Italian hospital, or whether they will want to run their own tests - which would essentially mean starting again. I have heard of cases where a foreign diagnosis was not accepted. This may be partly to cover themselves - if they accept a foreign diagnosis which later turns out to be wrong, they could be open to claims of negligence for not checking.
Usually, in the UK, there is not a second six-month scan to talk about treatment. IF the MS has been active enough (usually at least two relapses in two years) a diagnosed person would be considered for treatment straight away.
I had a six-month scan before diagnosis, but was to help with diagnosis, not to decide yes or no for treatment. I have never had another scan after diagnosis, and this is not normal here, although it is sometimes used to check if a treatment is working, or if the disease is active enough to justify a more powerful treatment.