Many thanks! I was not really expecting any answers but this is great! I vaguely remember several rows of seats, so Rapide or Dragonfly would be possible. My mother remembers this as being a pleasure flight, my memory is that we went to BA on a propeller plane, and I only remember one flight in a prop/tail dragger. This could have been to catch the Comet.
I had always assumed that the Comet was a BOAC one, but a quick look on Google shows that AA's had wing tanks and silver/blue /white livery on the plane and the tanks, exactly what I remember, because at 4 y o I really wanted to ride in the tanks! I have a vague memory of BOAC written on the tanks, but that could be a "supplanted" later image. We stopped for fuel in Africa, possibly Dakar. We were in a terminal restaurant and our departure time was suddenly brought forward as a matter of urgency, and we had to run to the plane. As we reached the steps I realised I had left my sausage roll behind and ran off across the apron to collect it. I was greatly amused by absolute chaos as countless people and vehicles went into panic mode to catch me!
I will ask my mother if the name Bill seems right.
Very grateful to both of you!