Short answer: nearly no effect on the service.
For some cases, airlines merge means that only 1 airline brand exists. United & Continental, American and US airways are the examples. The aircraft, reservation and all others things are combined into same brand names.
On the other hand, some airlines will keep the subsidiaries' brand even when they totally take over another airlines, like Dragon Air (Cathay's subsidiary), IAG group (Parent company of BA & Iberia) and Air France-KLM. Although they have different brand name but they may combine their reserve system and other equipment, they may also codeshare their flight.
As the CEO of IAG said in May "Aer Lingus would maintain control of its brand and operation while gaining strength as part of a profitable and sustainable airline group in an industry that’s consolidating", Aer Lingus will maintain its own brandname even after it has been takeovered. Moreover, the part of the deal between IAG and Irsh government (25% Aer Lingus share) required not to decrease services between London and Ireland + North Ireland and increase transatlantic services.
Given the fact that BA wont forgo Aer Lingus brandname, and they wont reduce Aer Lingus transatlantic services in near future, plus the fact that BA wont offerfly between Dulles-Dublin flight, the services wont be affected in the nearly future.
PS: British AirwaysBritish Airways is not in the process of buyingnot in the process of buying Aer Lingus, IAG has already boughtIAG has already bought Aer Lingus (98.05% share)