I'm reading a newspaper from 1907 in which a man says the following:
"I recall somebody on a wheel just going by me and almost at the same moment I heard light footsteps and a voice exclaimed, 'Good morning.'"
Given the era and the fact that the passage in question referred to a well-to-do part of town around 9 in the morning, I believe the man "on a wheel" was riding a bicycle to work. Does this interpretation of the phrase make sense?
I have seen this phrase used in early French and British sporting journals of the era, as you intuit, with regards to people riding bicycles, though usually with regard to the "penny Farthing" bicycle, with one very large wheel and one small rear one. However, the penny farthings were no longer being produced by the late 1880s, having been replaced by the more modern ones we know today with pneumatic tires. So, in the case of your 1907 paper, I would suggest this is a continuing use of an older term, and probably as you guessed with reference to the newer type of bike.