The soldier describes himself as 'private first class'. There is no such rank in the British army.
The cycle that Laurie piggybacks on with his mother has a very modern brake lever, probably from a mountain bike, and cable brakes. At the time the film is set, the brakes would most likely have been connected to the levers by rods.
When Loll is at school and his form says the Lord's Prayer, they begin 'Our Father who...' but in the 1920s the older version 'Our Father which...' would have been used. 'Our Father who...' did not become common in England until the 1970s.