Customs

Assuming that you are on one ticket and your Italy flight is from Heathrow also (some leave from Gatwick), you will not go through immigration in London. When you arrive in London, you will stay airside in the secure area. You will follow the signs for flight connections. You will need to see if your flight to Rome requires a change of terminals which means you get on a bus to go to another terminal. You will go through a security check. In Rome, you will go though immigration. The Heathrow website has good info on flight connections.

Customs and immigration are two different things. I've never flown through London or into Rome but have connected in Paris and Amsterdam. In my experience, since you have entered the EU, you will go through immigration in that stopover. When you get to Italy and claim your bag you will be subject to customs. I've only flown into Bologna from the states but have never had to go through customs. Rome could be quite different.

I'm afraid Ron is mistaken. You will not go through immigration in LHR if you are continuing on to Rome. Even through the UK is part of the EU, it is not a party to the Schengen Agreement.

Yup, my bad!!