The general point about codeshares applies here, namely that subject to all the other factors, the core allowance is Aer Lingus' rules, indeed when booking on BA.com there is a clear note that effectively says that EI's rules apply.

So if you have a LHR-SNN-LHR standalone ticket then you may have merely EI's allowance, which could be zero or one bag depending on the ticket type you bought. You can buy additional baggage allowance by telephoning EI in advance, it's usually cheaper than doing it at the airport but some FTers have reported problems doing this with BA codeshares (though equally some have managed it without issue). Moreover I suspect you will have to recheck your bag in Terminal T2 rather than having it through checked, which would require one ticket rather than 2, and that also means you need to get to T2 via the landside route.

EI are fairly tolerant about bag size, it is somewhat over the limit but LHR-SNN is rarely a full flight in my not-huge experience, so you may be OK. At check-in in T2 and SNN, I would have the backpack on your shoulders rather than presenting two hand baggage items in clear view.