Well, you ought to be okay, but it could easily be a bit uncomfortable. If it were me I'd phone AA and explain the situation, and see if there's a later departure from T5 that you could take. (Where from, where to?)

Otherwise, you'll need to go from the AA gate in Terminal 3 to the "flight connections" bus stop, where you'll be put on a bus to T5. When you get to T5 you'll go through a documents check, then upstairs to a security point for the usual x-rays etc. From there you're released into the T5 shopping mall; the departures level is down one escalator/elevator run from the security check point.

Most of AA's gates at T3 are about as far from the flight connections bus as they can be - it can be quite a schlep. I don't recall that Heathrow has any electric carts to expedite things, which is all the more reason to inquire about a later flight.

I've done a couple of T3 < > T5 transfers in the past month or two, and the longest it's taken was around 70 min. end-to-end. But that was riding in business class, which meant I was first off the plane; if you're in the back, or given your need to fetch your stroller, I don't think you could be that quick. And, of course, late arrivals at Heathrow are becoming the norm.

Good luck and let us know how it goes.

As Gardyloo suggests, you may be able to arrange for a later connecting flight, depending on your final destination.

If you stay with the original flight, contact AA and ask them to arrange assistance for you. They can have you met on arrival by someone with an electric cart.

There is a well-organized and well-used system in place at T5 to speed people needing assistance (mainly elderly people and disabled, but also people with young children and babies) around the terminal and I assume there must be something similar at T3.

The carts aren't usually 'visible' to most people scurrying through the airport because they use special elevators and wide corridors below the concourses and shopping levels. It's a pretty amazing experience to be whisked around 'behind the scenes, as it were, which I had an unwelcome opportunity to experience first-hand two years ago when I injured a hip while in Italy and couldn't manage the long walks necessary to board my flight back to the US.

You probably wouldn't be able to transfer between terminals by cart, but will probably use the bus normally used for facilitating flight connections between terminals. Then you should be able to arrange to have a cart meet you when you arrive at T5 and take you to your onward flight.

Here is a link detailing assistance available at Heathrow: http://www.heathrowairport.com/heathrow-airport-guide/special-assistance/request-special-assistance

Given that Heathrow handles millions of connecting passengers who don't have English as any language the best advice is just follow the signs to "Fligth Connections" and act dumb. The folk there will get you to your correct terminal.

The really good news is no immigration queues.

Thank you all for the advice.
We are coming from Dallas in The States, and heading to FCO (Rome). I will look into any later connections from LHR to FCO in case we don't make our flight. I hate to have an extensive layover with a tired toddler in tow, so I am hesitant to go ahead and change our flights... but I also know that if we miss, there is no guarantee that they would have room for us on a later flight. I'll keep pondering on this one. Thanks again for the detailed instructions!!

Be sure to ask the flight attendants if your names are on the list for the electric cart when you first board the aircraft to LHR.

Back in the fall I sprained my ankle before a trip to Europe and thought I might need assistance in LHR. I called the airline to request it (Delta in my case), made sure it showed up in my reservation online, and even re-confirmed it when I checked in for the flight in the US.

When I arrived in London, I was told my name wasn't on their list for people needing assistance! The one cart there was full, so the people at the gate told me I could wait for someone to come get me, but eventually I just gave up and walked to immigration.

I don't know if speaking to a flight attendant when I boarded would have done any good, but I think they usually have a list of who needs assistance upon arrival.

To update, we arrived late at LHR on our initial fight. A kind airport worker gave us a pass that allowed us to pass through security quickly. We made our connection, but the airplane was already boarding when we arrived at the gate so we didn't make it with a lot of time to spare.
The pass was bright orange and the person who gave it to me was staffing a security checkpoint. I showed her our tickets asking if she thought we should try to make the flight or start calling the airline about a change, and that is when she handed me the pass and said we ought to make it if we used it.