Deliberate mistake: Paddington puts a red sock in the load of prison clothes which turns all their clothes pink. However, it was a single red sock in the "uniforms" machine, but in the next scene the entire prison is wearing pink clothes now. The machine is nowhere near big enough to wash all the uniforms at once - the pile we see is huge.
Continuity mistake: Paddington gives chase on 'dog back' to the thief after he has stolen the book from the antiques shop, catches up with him, but the thief drops a smoke bomb and disappears. The dog is in shot as Paddington talks to the thief, but when the smoke bomb is dropped we get an aerial shot of the whole area and there's no sign of the dog.