I just did on Wed a T4 to T1 transfer from JFK onto FCO although all on BA - from the time i got on the T4 bus to the BA F lounge in T1 was about 30 minutes but everything flowed smoothly.

The key is when you get to T1 if you are flying J or F on BA there is a special line - however it is not well marked - the only sign you will see is at the entry to the lan which when you enter the actual area where they screen your bags and you walk through the metail detectors will be straight ahead on the far side of the hall when you turn the corner -

When there is a HUGE line for the non BA F/J screening those lines back up and thus block access to the BA priority line and you can literally spend 30-60 minutes waiting in that line and then find out the priority line is empty.

My advice is to stay close to the near wall - closest to where you came up so you can bypass the folks not waiting for the BA F-J line - just keep your F or J BA ticket out and act like you know where you are going so people know you are not trying to cut their line. Eventually the sea of people will part and you will be off to the races!