Have to say sounds like the crew was not great but I do wonder if the Op was looking for issues travelling with a baby in First and maybe read some of the comments on here! I do not think the crew should do anything special as you are travelling with a baby but they should give the service you expect in F and that did not appear to happen here.

Neither BA nor the OP covered themselves in glory on this flight!

Personally I am not sure about the mobile phone playing nursery rhymes myself, I really don't think the F cabin (or any cabin on a plane)is the place for the speakerphone function of your mobile!