I've never flown on any airline with a baby so my experience is somewhat limited at the moment I will say though that one possible advantage of BA over EasyJet is being able to check in online and thus minimise the amount of time in the airport.
That's probably less relevant with a baby than without, since you're v likely to get preboarded on EasyJet so no real need to turn up 2 hours before the flight, but worth a thought. My intention when I do fly with the bean is to go from Norwich as it's fantastically quiet, and you can turn up with absolutely minimal time if you are already checked in online. (Plus if you ask nicely the security people will post your letters for you if you've forgotten there's no postbox in departures!)
BA are generally less likely to be delayed than EasyJet, which also might be a factor to consider.
Btw, you say you are flying from Edinburgh "via" London, I assume you realise that EasyJet don't do connecting flights, so if you are travelling onwards you will need to pick up the bags in Stansted/Luton and then check in again? Plus leave a fairly good chunk of time between flights as you're effectively on two separate trips.
A friend got caught out flying from Edinburgh on EasyJet by the fact they only broadcast the announcements in half the airport - the half nearest the domestic gates, admittedly, but the other end, where the international flights go from, is much quieter and less smoky so she was there with her 18-month-old during a rather lengthy delay. She missed the announcement that the flight had moved forward by about 10 mins and turned up at the gate to find out the flight had closed and all her waiting had been in vain. Just a little one to watch out for!