Transfer time at LHR (T3 → T2 → SNN)

Since the OP has already booked tickets, I don't think any of the information I'm adding here is going to help much, but just to set the record straight.

According to expertflyer, AA and EI do have ticketing and baggage interlining agreements. I can bring up a fare on matrix for ORD-AA-LHR-EI-SNN on a single AA fare which would have allowed through check of baggage to SNN.

Having an interline agreement doesn't mean that baggage must be interlined on separate tickets though, as that is down to the policies of the airlines. AA's policy appears to currently only interline to other oneworld airlines on separate tickets and even that is likely to change soon due to a recent oneworld policy change.

If you are following the same itinerary in reverse on the way back, Aer Lingus also state they will only interline bags on the same ticket:

Aer Lingus will through-check baggage as a courtesy onto any carrier that have a baggage agreement in place, upon presentation of documentation of the itinerary showing Aer Lingus and partner carrier flights on the same ticket.

This includes the following airlines: American Airlines, Air France, Air India, Alaska Airlines, Finnair, British Airways, Air China, China Airlines, Cathay Pacific, China Southern, Emirates, Etihad, Gulf Air, Iberia, Japan Airlines, KLM, Kenya Airways, Kuwait Airways, Malev, Malaysian, Martin Air, Northwest, Austrian Airlines, Pakistan Airlines, Qantas, Qatar Airways, South African Airways, Singapore Airlines, Thai Airlines, United Airlines, Sri Lankan Airlines, US Airways, Virgin Atlantic and JetBlue Airways.

Although Zach's references indicate that passengers travelling on separate tickets should not use flight connections, if you don't have checked baggage I see no reason why you should not use it. Your previous flights are of no importance, only your boarding pass for your next flight. If you have checked bags, that point is moot if your bags aren't interlined.

The fact that you have purchased two separate tickets (as in two unrelated transactions and neither airline knows about the other flight) will make things substantially more difficult.

If you bought the flights separately, you officially can't use the Heathrow Connections service and will have to exit the plane and walk to arrivals, go through UK immigration (assuming you have the documentation needed to enter the UK, which is straightforward if you're a US Citizen, but may be less so otherwise), claim any checked luggage, take it through customs, get yourself to T2, check in any checked luggage, go through security, get to your gate, and board. As a practical matter, it appears that you could do so anyway if you can produce the boarding pass for your second flight (see comments below), but none of that helps you if you have checked bags.

The walk between T3 and T2 is about 10 minutes and fairly straightforward.

You'd need to get to the Aer Lingus check-in counter 45 minutes before the departure time to check-in your bags. If you have no checked baggage, you may possibly be able to check-in online, which would save time. Officially, you must be at the gate no later than the time on your boarding pass, which should be 30 minutes prior to departure. They will close the flight, if it is on-time, sometime before the departure time and will not allow you to board, even if the aircraft is still there.

I would say that this is not enough time. It could theoretically work if there are no lines at immigration or security and your bags come up quickly. It could easily fail if the AA flight is a bit late or there's a 45 minute wait for immigration. And if it doesn't work, Aer Lingus has no obligation to assist you, as you arranged the connection yourself and they aren't guaranteeing it.