It's not very clear from the movie -- those uniforms could also be Canadian WWI army uniforms -- but I think it's most likely he's fighting for the US. The evidence is very circumstantial, but Logan also fought in the US Civil War. This means that he fought in the Civil War, WW II, and Vietnam war for the US Army, so the simplest explanation is that he simply relocated permanently to the US and fought for them. He only moved back to Canada after he abandons Task Force X, in the 1970's.
For completeness, I should point out that in the comic source (Wolverine: Origins), Logan definitely fought for Canada in World War I.
However, his comic book past has been retconned about a dozen times -- in-universe, most of what he thinks is his past is memory implants. There have two Origins mini-series and a full Wolverine: Origins series just convering Wolverine's back story. He also moves around a lot in the comics: he was in Canada in 1907, part of the Canadian army in 1912, he was in Mexico in 1920, Madripoor (fictional, but close to Singapore) in 1932, etc. I'm not sure it's even possible for all of those stories to be simultaneously correct.
So, it's probably not productive to look to the comics material for any specific details about the movie universe.