As a child my sister and I went abroad several times with various aunts and uncles and grandparents, we never had any issue to my knowledge. Perhaps for the reason of completeness my Mum may have written some kind of letter that one of the adults we were with carried. I've never been abroad with my Dad (parents divorced when I was young) and I don't think my mum ever got my dad to sign a letter saying she could take us out of the country. But then again I'm going back over 15 years, times have changed. The first few times I remember going abroad I was just a name on the observations page in my mum's passport and as a teenager I went away with school on a group passport.

As it's a domestic journey and granny is travelling I for see it being a problem for you, but confirming this with BA is worth the peace of mind.