So you are flying Phoenix - London Heathrow - Rome. The PHX flight is on AA and the Rome flight is on BA. Do I have that right?
As far as I know, there is only one PHX-LHR AA flight a day, AA 194. I was on that flight last November. It departed from Sky Harbor Terminal B. EDITED TO ADD: I saw later you are on a BA flight PHX to LHR. You should arrive and depart from T5.
I fly LAX-LHR-Europe every year. I am familiar with the procedure. I usually fly AA LAX-LHR then BA LHR-Europe. (Last year flying from PHX was an exception.)
You will not have to worry about checked bags in London. Assuming you are on one ticket, your bags in PHX will be checked through to FCO. I think you might be mistaken about your need to collect bags last year when you went to CPH. Either that, or you were on two separate tickets.
I, personally, don't find it such a nightmare as some describe. I do it every year and, yes, I am a senior with creaky knees.
Here is, in detail, what you should expect.
1. Check in at PHX, and get your luggage checked all the way to FCO. Confirm with the check-in agent what terminals you will arrive in and depart from in London. Ask for a BA LHR-FCO boarding pass. In my experience sometimes I get it, sometimes not. If not, then I will have to get the LHR-Europe BA boarding pass in London. Not a big deal. It's just slight more convenient to get it at check-in.
2. Board the aircraft and fly to London.
3. Arrive in London at (most likely) Terminal 3. You will depart (most likely) out of Terminal 5. After your PHX check-in, you will know for sure.
4. When you get off the plane in London, follow the purple Flight Connection signs. You can't go wrong here. If, for some reason you arrive and depart out of the same terminal (not likely in my opinion), you still follow the purple Flight Connection signs,. If you arrive in T3 and depart from T5 (most likely) you still follow the purple Flight Connection signs.
5. Flight Connection purple signs will take you to a departing bus station if you are changing terminals. Just look for the Terminal 5 bus line and wait there. It should be a short wait. There are chairs available should you, like me, need to sit down.
6. Once in the bus, if you are in need of a seat (as I am) grab one. If you are one of the last to board and fear the seats might all be taken, then just wait for the next bus. I've done that, no problem. They have a few seats with blue handicapped reserved stickers. If an able-bodied person is seated there, just give them the "Grandma Look of Where Are Your Manners?" and hope they volunteer to move.
7. The bus takes you to your departure terminal, most likely T5.
8. This is important to understand: All connecting passengers (excepting I believe people connecting from UK airports, but I am not sure) MUST go through LHR security. From reading your message, I think you were not expecting this last time, as you might be accustomed to USA connecting airports where passengers go through TSA only at the beginning of their trip. In London, you and everyone else will have to go through their security.
9. As you are flying Business Class, you are eligible for their "Fast Track", so it's a bit easier.
10. After you get off the bus, follow again the purple Flight Connection signs. You will find yourself in a large area where there are airline check in desks and lots of LHR employees to help guide the befuddled passengers. This is the beginning of the security process. If you need to get a boarding pass for your BA flight, do so here.
11. With boarding pass and passports in hand, look for the Fast Track line. You will first show boarding passes and passports to an agent who confirms you have arrived in T5. This is (I believe) called Flight Conformance. Others may correct me if I am mistaken. Go past the Flight Conformance podium, follow the crowd and go up an escalator.
12. Get off the escalator and follow the signs to Fast Track Security.
13. The LHR security process is a bit different than US TSA security. You don't have to remove your shoes, unless they are heavy-looking and/or very metallic. I think they are only looking at shoes with a thick sole where contraband could be hidden. (Tip: Don't wear heavy shoes on your flight.) You do have to remove all electronic devices: laptops, tablet computers (iPads, Kindles, etc), any bundles of charging cables, wires, etc. They go in the plastic bins. Similarly, your 3-1-1 bag of less than 100 ml (approx 3 oz) toiletries go in the bin as well as overcoats, etc. Go through the detector, the collect your personal belongings at the other side.
Remember: Your Global Entry or TSA Pre-Check status means nothing here. You have to follow their security rules. Essentially the only break you get is a shorter line in Business/First Fast Track. But you get to keep your shoes on.
14. After leaving Security, you will find yourself in the main departure lounge. You wait here until your gate is announced. LHR, unlike most American airports, do not announce a gate until they are ready to take passengers at that gate. You can't just go to your gate and sit there for a hour or two. With your AA Business Class ticket you have several options for waiting.
15. First, look at the monitors to see what they say about your departing flight. If your flight will depart from some still-unknown gate in Terminal B, for example, you can go to Terminal B and wait there. There is a BA lounge in T5, I believe. If the monitor says nothing about Terminal B or C, then you will depart from the main Terminal A where you are now.
16. There are two BA lounges in Terminal 5 open to AA Business Class travelers, one at the north end and the other at the south end. In a perfect world you would know where your departure gate is, and pick the lounge closer to your departure gate. Alas, not possible here. So you go to either the north or south "Galleries" lounge and wait there. Your other alternative, and it's not a bad idea, is to just pick a comfy place to sit in the main lounge about midway, thus avoiding a long walk from (eg) the North lounge to a South gate, or vice versa. Pick a place where you can easily watch the monitors. I do that when I have a relatively short-ish wait and don't want the long schlepp to one of the lounges and then, with my luck, an even longer schlepp from lounge to gate.
17. Watch the monitors and when they say "Go to Gate XXX" for you flight, get up and go. Slower walkers should not dilly-dally. Fast walkers could take a few minutes for the restroom, etc, but a slow walker such as myself should get up right away and go directly to the departure gate.
18. Wait at the gate and board the aircraft.
19. Arrive in Rome. Here you will go through the Immigration and Customs formalities. Stand in line to show you passport to the officials, answer questions and get your entry stamp. Then follow the signs to the baggage carousel and get your bags. Carry them through Customs. Unless you have something you need to declare (excess cigarettes or alcohol, etc), then just go through the green Nothing To Declare door and out to the main Arrivals area. This is where you could be met by your cruise line people, or make your own way.
Final note: In Europe, restrooms are "toilets". Weird to us Americans, yes I now, but over there it's totally normal and polite to ask for the "toilet". Signage in the airport and other public places will say "Toilets". In Heathrow, for some crazy reason, the door to "Toilets" usually takes you into a long, hallway with a few left and right turns. Just keep going, assured there will be restrooms at the end of the trek. Be glad there are no stairs. In many public buildings in Europe, sadistic architects put the restrooms in the BASEMENT, requiring trips down and up stairs. Pro tip: Be proactive.