The Field, a British country gentleman's magazine, began covering golf regularly sometime in the early 1870's. UK newspapers began covering golf 'fixtures' at about the same time. By the mid/late 1880's there were a number of other UK periodicals with columns on golf. The first Golfing Annual was published in 1888 in London. I'm not sure when The Times started its golf column, but I'd guess it wasn't long after that.
It seems reasonable to think that a fair number of Americans, even if they had not traveled to the UK, read those magazines at the time. The Field, for example, had a diverse readership. It focused on the whole spectrum of 'country' interests, from raising cattle and chickens to building walls and everything in between. So readers who bought the magazine for pointers on managing their properties would have also run across articles on golf on a regular basis.
I have always thought that the rise of popular journals like The Field were part of the story of the growth of golf, even in the US.
Bob