Not directly applicable, but when doing the T3 <-> T4 transfer plenty of times, I've often found the connection center completely stuffed with people, so it's often faster to just clear immigration and re-enter. Doubtful that's the case with T1, but keep it in mind as plan B. The AA lounges are just where the escalator start, and they are both crap. There's also a nicer CX lounge in T3, at least there was one a few years ago.

Also, watch out for bag nazis, last time I transferred through LHR I tried several different security lines before I was allowed to pass with two carry-on pieces. These days, Italian airports appear well-thought out and efficient in comparison.