BA (nor EI) won't be forced to give up LHR slots by default.
What will probably happen (based on precedent set by IAG's purchase of bmi) is BA/EI will have to surrender a set number of slots (say six pairs) to any willing entrant on LHR-DUB (ditto for Belfast).
Based on the experience of Little Red, it's unlikely that an entrant will be forthcoming.
I don't buy the idea that IAG will slash connectivity between Ireland and the UK.
One of the first things IAG will do is, as soon as regulatory approval allows, put the EI code on every BA long-haul route. And vice versa. Ditto for Iberia at MAD.
Also don't forget that EI will gain a very strong transatlantic partner in the form of AA so I expect AA will increase its flights to DUB above existing services. AA also has a huge hub at MIA and DFW (and to a lesser extent LAX) so there's plenty of scope for EI to add destinations in the US.