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Is it a good idea to apply for a British passport?
This depends on whether you want to travel to the UK (especially for long periods of time) after it leaves the freedom of movement regime, or to another jurisdiction where British citizens enjoy fewer requirements than Italian citizens. A cursory check of Wikipedia suggests that no such jurisdiction exists, however; see below for details.
Am I still eligible?
Almost certainly. You can only be deprived of British citizenship involuntarily in very limited circumstances. For most people, the only way to lose it is to renounce it formally.
If I qualify, what would be the quickest and easiest way to obtain a British passport while I am in Italy?
See Lost UK Passport on this site.
What advantages are there in having a British passport vs. an Italian one?
You can prove your right of abode in the UK with one and not the other. Depending on UK immigration requirements for EU citizens after the UK leaves the EU, and on your travel plans, this may or may not be significant to you.
A quick comparison of Wikipedia's articles on visa requirements for British and Italian citizens suggests that the only differences are in Turkey, Iran, and the United Arab Emirates; in each case, the requirements are more restrictive for British citizens. It's possible that some countries' visa policies for British citizens depend on the UK being in the EU, so more differences might arise after Brexit. But it seems likely that any changes would be more restrictive for British citizens rather than less.