You should visit an Ontario government web site that has a searchable index of the 1871 Canada census. I'm sorry I do not have the URL as I lost it in my last computer crash but it probably could be easily found by typing "1871 census Ontario" at google.com. I found many Shaw's listed there including from the Ontario region. In my search, I had determined that London area Shaws were not related to my line as they were not French speaking.
Good luck in your search!