#huzzay

"You Say Huzzah! They Said Huzzay!" by Norman Fuss at the Journal of the American Revolution

Basically it argues - credibly, in my opinion - that the predominant pronunciation of the word in the English speaking world was with a hard "ay" sound.

As he points out in the article, it is nearly unilaterally spelled simply "huzza" in the period, in both formal and informal writing. Which can technically be read multiple ways, but I think his evidence is clear.

There's some scholarly debate in the comments, including at least one instance of a poem that seems more likely to be rhyming it more with an "aw" sound, so it's possible it wasn't universal, and there are also some debates (not on the article itself) that possibly different dialects and/or people for whom English was not their first language may have been a factor. Occasionally you'll come across some weird phonetic spelling, like some New Englanders spelling it "whosais", but still, I think he makes a strong argument at least for English speakers "Huzzay" was standard.