Next Spring we will be traveling to LHR and Landing at Terminal 3 on Virgin. How long will it take me to get to Terminal 4 and British Airways with NO checked baggage (only hand baggage)? Would we have to go to immigration?
The London Heathrow website includes a planner where you can work out step-by-step connections. http://www.heathrowairport.com/
It would be easier to sort out your immigration question if you indicate where you will be flying to on BA.
We would be flying BA to Rome and our planned layover is 2 hours 20 minutes.
Sorry, Alitalia to Rome.
No need to go through immigration. Follow flight connections, take the connection bus to T4, go through security. The MCT on a single ticket is 90 minutes.
The full URL for the Heathrow connection planner that Southam mentioned is http://www.heathrowairport.com/heathrow-airport-guide/flight-connections/connection-guides. Just put in all your particulars, and you'll get the answers to your questions.