During the chase in Hong Kong, when Johnny English walks off the pallet after having been craned between buildings, he says Thank You to a builder in Mandarin. His experience should have told him to use Cantonese instead.
When Johnny is in the restaurant with Agent 1, he drains his glass of wine and agent 1 asks the waitress for another bottle, in the next shot there is a small amount in the bottom of the glass, yet moments later the glass is about a third full.
During the aerial shot before the MI7 meeting in London, the wording to the bottom right of the "camera" screen says "GEOCACHING". Geocaching is a hobby using GPS to find hidden items, not something which should be on this sort of camera readout.
When English visits Tucker at his home, and Tucker is playing a racing game on his laptop, the movements of his car in-game seem to have nothing to do with his actions on the keyboard. When he then stands to close the curtains and talk to English, the car continues to behave as if being controlled, showing that it's just a demo video.
The goof item below may give away important plot points.
Near the end of the film when Johnny English goes into spasm, he drops his pistol. Briefly, you can see stamped on the side 'Tokyo Marui', 'made in Japan'. This is a replica BB gun, and fairly harmless.