The victim's daughter climbs into the family's Auster plane and prepares to fly off. Father Brown joins her and she takes off, but the propeller is stationary when both board the plane -- and the engine must be started from the outside, by twisting the propeller.
De Sacramento Extremae Unctionis (Last Anointing) is not given to a person already dead, and it is done on five parts of the body: eyes, ears, nose, mouth, and palms of the hands (unless the person is a priest, then the backs of the hands). Anointing the feet also, is optional, but there was no reason to anoint the thighs since he still had feet.
Children are not allowed to fly planes. Even in the 50s.
A 13 year old piloting an aeroplane warrants more than an angry conversation with their passenger and being sent to their room.
When the plane takes to the sky vapour trails can clearly be seen in the distance.
When the inspector is going to bring in Geoffrey for questioning, Millie is told by her mother to go to her room. We just saw in the previous scenes that the murder victim was killed in Millie's room, and we have no reason to believe the body has been moved, or the crime scene released by police. Even if the mother would say such a horrendous thing, we must believe Father Brown would have said something to stop it.