I don't live in London, but I find that dealing with the Italian consulate in San Francisco to be very fast. You have to make an appointment (usually a couple of weeks out) but it takes 20 min. to take the fingerprints with the new passport and to generate your passport, but when you are done you walk out with your Italian passport. I can't imagine that the process at the London's consulate is any different.
To go to Italy you don't need a valid Italian passport, because you can re-enter Italy with an expired one, and to go back to the UK you can use your British passport.
Since you are a resident in the UK, the Italian Consulate in the UK that has a jurisdiction over your town and where you should be registered as an Italian living abroad, is the one that can issue a passport for you.
In particular cases other issuing offices, such as another consulate (other than yours) or a questura (police headquarters in Italy, normally competent for issuing passports to Italian residents in Italy) can issue a passport for you. But if you don't request the passport from the issuing office with jurisdiction over where you reside, but you go to a Questura in Italy, they will need a written authorization by your consular office (where you reside) authorizing the Questura in Italy where you apply to issue your passport on their behalf. So where is the time saving? You still need to deal with your Italian Consulate in the UK one way or another.
You could go to the Questura and say you lost your passport while in Italy. In that case, if you need it right away, they would give you a temporary passport, which doesn't require fingerprints but that is valid only one year. So in the end you would still need to deal with the Italian Consulate in the UK sooner or later to have the full electronic chip passport that Italy issues nowadays.
So my suggestion is to go through the proper channels and get your Italian passport renewed in the UK. Chances are that it will take less time than you think.