Of course, Jehu is pronounced JEE hyoo or JEE-hoo in the traditional U.S. pronunciation, which is the one I am familiar with from my childhood, when I attended Sunday school regularly and, at least until my teens, used either the King James' or Revised Standard version of the Bible (which conveniently indication the "proper" pronunciation of such names). The assumption that this name is pronounced JAY-hoo, which thedayhascome put forward and qroqqa corrected, derives, I suspect, not from any Italianization or Gallicization or Latinization of the name, but rather from its similarity to the name Jesu, an archaic or poetic German version of the name Jesus and which is pronounced, more or less, YAY-zoo, though a lot of native English speakers pronouce the name JAY-soo, especially when they see it in the English title of Bach's chorale, "Jesu, Joy of Man's Desiring," primarily for euphonic reasons, i.e. to preserve the alliteration of the song ("Jesu - joy"). So if someone knows the name "Jesu" from the Bach chorale, and then encounters the unfamiliar name "Jehu", it's not unreasonable (though incorrect) to assume the latter is similarly pronounced.
Also, the answer to qroqqa's question, "When did people i.e. English speakers start Italianizing Hebrew names?" is "Never, really," although as seanahan's comment suggests, no Hebrew names, from the Bible at least, came into the English language without first going through Greek and Latin, e.g., Matthew, James, John, Mary, Jeremy, David, Jonathan, etc. – all these names would have come into English via the Vulgate (the Latin Bible).
Of course Italian isn't Latin, but once upon a time it was - the so-called volgare - so I don't have any problem with the way seanahan put it. I think you're a bit flattering about how large French/Italian loom on our horizons: most English speakers would be familiar with Reykjavik for example. J does not occur naturally in Italian. About the only offering in current use is the football club name Juventus, happily pronounced with /dʒ/ by most non-Italians anyway.
No, Italian is not Latin, though admittedly it would save a lot of etymological shoe leather if we called them all Eurasian. The Italianization in question is the supposition that all foreign languages are more or less the same up to some pesky isomorphism, so pronouncing them like Italian/French is good enough.