Hello,
My family is visiting Europe in a couple weeks and booked plane tickets to London Heathrow a few months back, but have recently decided that we want to fly directly to Dublin when we arrive at the airport. We are supposed to arrive at LHR at 7:05 with British Airways/American and the flight to Dublin is at 9:50 on Aer Lingus. We are checking bags and want to know if this is possible or should we take a later flight? There are also flights at 10:50 or 12:15 for a little bit more money.

Thanks

What is the arrival terminal?
What is the departing terminal?

Front of the plane or coach makes a HUGE difference but you almost certainly arrive a T-5 and depart T-2. I sure wouldn't on unprotected tix. Business/first I'd do it because you'd get expedited immigration, but would still have to wait for luggage, but if you are in coach no way no how. If in coach, I'd take the 12:15.

Traveling carry on only would help too . . .

Arrive in T5 departing T2

Yes we are all in coach. So a 3ish hour transit wouldn't work?

Could 3 hours work? Sure it 'could'. But many long haul flights arrive in the early AM and immigration queues for the poor souls in coach can be ENORMOUS. If you miss the connection you have to pay full, walk up fares for new tix . . . and there may be no seats available for several flights. Do you feel lucky?
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If possible I'd re-think things so you have some time in the UK before jetting off to Ireland

Add in possible winter weather delays, too. Heathrow is HUGE, I wouldn't be comfortable choosing anything but the 12:15 either.

Expedited immigration for first & business class simply means you can use the Fast Track Arrivals lane, it is a separate line but might still have to wait in line, with all the other first/business passengers. Arrivals at T5 sometimes require you to take the train to get to arrivals which takes time. Arriving at 0705 and taking a 1215 flights seems far too big a gap no matter which class you are travelling, considering most flights arrive early from the US. 1050 is more comfortable IMO than 0950 Even better, fly LHRDUB on BA from T5, although BA will not thru check luggage on separate tickets.