Neale tells Rennit he is cross that his hotel room has been ransacked. That scene does not appear in the edit I saw - he chases the blind man then the next scene is at the investigator's office.
When Neale gets off the train he leaves everything he has on board, including his hat. When he arrives at Rennit's office though, he has a hat.
The way that cake box was flung around and handled, the cake would have been crumbled and in clumps when he got it out to serve to the blind man on the train. At least, most of the frosting would have been stuck to the inside of the box.
The amount of rain on Milland's overcoat changes in the tailor shop between the long takes and the close-ups.