Very good point. The doctor at no time even suggests that he examine Iris' head.

By the way, considering the flower planter was pushed off a balcony that was on the second storey of a building, and was obviously extremely heavy as the person who pushed it struggled to get it to slide off, in reality I doubt Iris - had this been actual life - would have been able to even function throughout the remainder of the story as there would have been a definite serious injury, in fact near certainly major back-of-head trauma.

For real, she got brained by a porcelain flower planter from 20ft up and shook it off like it was a stray tennis ball. Maybe that was some deliberate use of absurdity? The film begins with pure comedy for the first 20 mins, so maybe the whole "bump on the head" subplot was intended to be a little silly while drawing us into the paranoid drama that soon follows. So it's easy to miss the fact that the doctor never examined her. But definitely keen audiences would notice a clue there.

Another clue is his accent. If you notice, the film makes very deliberate use of accents to label each character. The 2 posh guys are annoyingly clueless & useless (royalty?), while Mrs. Froy and the 2 protagonists have a more neutral accent that doesn't pigeonhole them and leaves their characters flexible. The cockney/manchester accent of the nun reveals her to be a working class type who gets things done (unlike the posh gents), and indeed she sorta saves the day twice. Of course the doctor in question has an ambiguous East European accent which was, in 1938, the enemy.