Marvellous stuff, I'm sure you'll agree. In fact, "The huntsmen are up in America" is a phrase I like so much that I sometimes catch myself saying it round about midnight. It's less infantile than "Up the wooden hill to Bedfordshire," after all. But more often than not I bite the words back - because, as a moment's thought will reveal, Browne (being sleepy) got the Earth's direction of spin wrong. By the time the huntsmen were actually up in America he would have been tucking into his elevenses and the Persians would have been taking afternoon sherbet.
I've considered adapting the phrase to reflect geographical reality. There are several suitable candidates that would preserve the dactylic charm of the original. "The huntsmen are up in Mongolia," for example. However, it's just not the same.
The only other expedient I can see is to move to a part of the world where Browne's phrase would actually make sense. If I lived in Honolulu, for example, saying "The huntsmen are up in America" at midnight would work perfectly, at least for the huntsmen of the east coast (whom Browne no doubt had in mind), while in Iran it would be the small hours of the morning - not ideal, but adequate. [ETA Actually the small hours of the afternoon, of course. Not so good.]
In fact, the more I think about it the more inevitable it seems that some future graduate student will use this phrase as the basis of an article arguing that Sir Thomas Browne was actually a native of Hawaii. I, for one, wish that person well.