According to my favourite online etymology (word origin) dictionary, the word doughnut was actually first used in 1809 in the US! It wasn't until 1929, according to a Wikipedia entry I found, that the spelling first appeared as donut. I would have to do some more reading on the history of doughnuts themselves, but if they did actually originate in the US, then the original spelling became popular throughout the British-English speaking world, while the spelling ended up evolving in the US to exclude the -ough.
Some people have equated the donut spelling with Dunkin' Donuts, but according to the same Wikipedia entry, which you'll have noticed if you clicked on the link, that restaurant wasn't founded until the 40s, meaning they may have popularised the spelling but weren't the first to use it publicly.
Apparently the spelling doughnut is as common in the US as is donut, but in my personal experience, I've rarely seen the original spelling in print.