Think Milano-Torino is the oldest, saw numerous tweets about it when it was ridden and none was corrected. If it wouldn't have been there would have been someone who corrected it.
Other pre 1900 races are Paris - Tours and Paris - Roubaix both from 1896. If I have to believe Wikipedia Roubaix has the most editions with 115
But what are the requirements of "race"? According to this article http://thebikeshow.net/the-worlds-oldest-bicycle-race/
The Catford Cycling Club ‘Hill Climb’ is from 1887 which would make it the second oldest, but if it's up to amateur races it's hard to figure out I think. (for example, as European I've never heard of the Melbourne to Warrnambool).