I am not a historian.

But, I think I can add something to this discussion, by using Google's Ngram Viewer.

I actually started this comment off differently, but after some Googling, I am now thoroughly confused as to whether "huzzah" of yesteryear was actually pronounced similar to what we would consider "huzzay," or if that's merely a variant. Google Ngrams suggests that, regardless of the pronunciation of the spelling, "huzza," the variation , "huzzay," did, in fact, exist in the 19th century.

I DIGRESS!

Take a look at this Google Ngram plot:

• huzza/huzzah/huzzay vs. hoora/hoorah/hurrah/hooray

You'll find that the transition huzz-type exclamations to hoor/hurr-type exclamations occurred sometime in the mid 19th century.

I would be remiss, though, if I didn't mention that the specific spellings of "hurrah" and "huzza" dominate that Ngram chart, though, so you can take a look at this alternate version of the chart with those two words removed, below:

• huzzah vs. huzzay vs. hoora vs. hoorah vs. hooray

Here, you can see the rise of, "hooray," in the beginning of the 19th century and the decline of all of the other variations at the end of the 19th century.