I don't believe that CW on BA is exactly the best place for very young children. Not exactly the easiest setup to keep an eye on them or gain access to them if they have their own seats.
One of my last CW flights had someone move to let a couple and their child occupy the centre seats. Made it easier to keep an eye on the child and provided a bit of floor space for he/she to play if need be.

Anyway, the parent + child seating should be done at the time of booking rather than at some point down the line, and especially not when people are boarding and things become musical chairs.

I've been fairly lucky child wise.
Had one woman and her young son next to me in bulkhead seats on a flight to SFO and he was fine.
Another time, an Indian couple, with a young lap child, were in the window and middle seats and they wanted to be with an elderly family member who was in the cabin behind. Luckily the crew sorted something out and they moved back to join her and a man moved forward and we ended up with a free middle seat so a better deal for all of us.

From above, it does appear that whilst on a plane and the little darlings are running wild, that not all parents have the notion that the kids are important



I've no real issues moving if it helps out someone, but it does depends on their attitude. Demanding I move / intimidation won't get me to shift, but politeness will (assuming I'm not going to get screwed over seat wise and end up in the middle of a row of 4 for example).