Ah - you didn't tell us about sisters! All clues are invaluable. There are indeed two female Bedworth births with mother's maiden name Smith in Pontefract, in 1932 and 1933 respectively.
Hmmm...
There are no Bedford births with mother's maiden name Smith in the area at the time, so I wonder whether John Bedworth's surname might indeed have been mis-recorded as Bedford at the time of marriage, or an error made in later transcription for the indexes?