When the BT telephone in the red phone box rings, it rings with an American (US) ring style. The phone rings with a long ring followed by a short pause then a long ring. In the U.K. the phone would have a double ring followed by a longer pause then a double ring. That all being said, the particular BT phone in the phone box wouldn't even ring with a bell, as heard in the film, but has an electronic ringer.
The second letter that Phoenix finds on the scavenger hunt is a "C", but when he is playing the notes on the organ to get to the treasure, the second letter he enters is "A" (he says "A" as he pushes the key). He should have played the "C" key.
The railway chase suffers from multiple errors: engine drivers have no control over where their train will go, even the most skilled of amateur drivers can't suddenly turn a family member into a burly stoker for a big hungry firebox, freight trains never commence their journey at platforms of London passenger termini, uncoupled vehicles will automatically brake to a halt, the couplings are American style instead of British, parallel tracks also have trains coming in the other direction, and buffers are very heavily built when a track ends next to a long drop into flowing water.
The goof item below may give away important plot points.
The police tell the Brown family Paddington was caught "red-handed" stealing the pop-up book, and he is convicted of the crime. Yet neither during the chase or his arrest is the book ever in his possession.