We flew Virgin last summer from London to Boston when our DD was only 7 weeks old. We booked her flight even before she was born. They have a policy whereby they designate the bulkhead seats based on the age of the children, so at 7 weeks old we were pretty much guaranteed the seats. They were very helpful all the way from the booking to helping with the SkyCots on the flight. They claim to have one flight attendant who is designated as the "family helper" but they were all helpful.
We flew BA at Christmas from London to Boston when DD was 7 months old. We were given the bulkhead for the outbound flight, but not the inbound flight. We tried calling to change the seats, but they said they had already been allocated to another family. The people on the phone were not clear on the policy for allocating the seats, was it first come first serve or based on age, etc. Everyone was very helpful at the airport and on the flight. The check in employee explained that she was tagging our fold up stroller in case it needed to be put in the baggage hold, but that if we asked at the gate they may put it on board if there was space. It pays to ask because the family in front of us had theirs taken away to the belly of the plane because someone at the gate (not a flight attendant) told them that was the policy. While he was busy taking the stroller away, we nicely asked the flight attendant if there was room on board. She took it right away. So we had it to go through immigration at the other end. I think it was the return journey, they actually opened the stroller up for us and it was waiting on the jetway, which is super helpful.
We are flying with BA from London to Phoenix later this month and the reason was that BA flies direct and we could use points to upgrade to business class. Now that DD is crawling and standing up, we thought more space would be better. But when she was little, economy (the bulkhead seats) was more than spacious on both airlines.