The is railway conductors hat.
What makes this a conductor hat rather than just a military peaked cap is the combination extra height of the crown combined with the peak pointed downwards.
The hat probably would have a had badge on the front with the name of the railway the owner or network it belonged to.
The pointing down peak perhaps was designed to intimidate or avoid eye contact while checking tickets, pure guesses on my part, similarly pointed down peaked are typical to military drill sergeants.
