"OK" can just indicate acknowledgement of what's been said, it doesn't always mean anything more.
If "OK" is all you say in response to "my wife is dead" then yes, that's rude in pretty much any context because more is expected of you than mere acknowledgement.
If all you say in response to "your wife is dead" is "OK" then it's not rude, it just means you don't have any other response to such shocking news.
If you say "that's OK" in response to "your wife is dead" then that does suggest that the death doesn't matter much to you (perhaps only only saying it because you're in denial, but still saying it). Although again, in such circumstances, you'd probably be held to lower standards of etiquette than normal.
Like J.R. says, using "OK" at the start of a sentence can be almost meaningless. If your employee says "my wife is dead" and you respond, "OK, that's terrible, I'm so sorry for your loss. You should take as much time off as you need.", then that's not rude and it certainly doesn't suggest in any way that the news is good. It's just a verbal tic, or possibly the employer trying to convey that the employee's wife's death doesn't present an immediate practical problem to the employer.
More generally than your specific case of bad news, in British English at least, "OK" is like "alright", it's often non-committal. "How was the movie": "It was OK", means it wasn't good. "How was the surgery?": "It was OK" might mean everything went perfectly well, but "OK" is as much as you're saying because surgery in itself is never good. "How are you doing?", "I'm OK" means nothing at all. If you don't convey anything by tone or expression then it's a non-answer.