I have visited Italy a lot because I was married for 8 years into the family of a man with an Italian mother.
There are many establishment in UK which describe themselves as Italian restaurants but which aren't.
There are several types of eating establishments, but the three main ones are - pizzaria, trattoria and ristorante.
To recognise the difference read this -
How Italians eat
I haven't dined out for about 5 years so I can't recommend any.
My adult son still loves eating out Italian, but I have never heard of him using any establishment which could be described 'local'.
So good luck with your quest, and although you may find some people may recommend a pizzeria masquerading as a ristorante, it just indicates their preference for the simpler food of pizza and pasta, which is not really what good Italian food is all about.