It's ok to use it in pretty much any email and (especially) phone conversation, even in business context. At least that's my experience from the software industry (communicating mostly with European and North American people). For formal letters and academic writing it would be a different story, of course.
But if you don't want to annoy people, please do not always write it in capital letters ("OK"), ok? :-) Maybe it's just me, but that seems somewhat archaic.
In live hare coursing, a bye means to pass to the next round due to a missing opponent, but not before the dog has made a lone run so it doesn't have a rest and an unfair advantage over the others.
The National Coursing Club's glossary says:
Sometimes runners are withdrawn from their courses, either because of absence, injury or weariness. Their opponent still has to run a course – a “bye” – so that it will have run the same number of courses as the next opponent. The dog may run alone or accompanied.
So in the first example above, greyhound Bismarck's opponent had pulled out, meaning he could progress to the next race, but not before having to run a punishing run to "even the playing field".
And in the second example, the day ends with greyhound Duffer running a bye on its own.
And again in the third example, where it's clear the opponent was injured the day before.
We can see bye used a lot in any of these books, and "run [...] bye" in many other coursing books. As can be seen, "get a bye" isn't in itself any kind of special set phrase (at least in these examples), but just some words used together.
Best Answer
MMM is in anticipation of "B".
I find descriptions of the use of "mmm-bye" going back several years and all across the country but most often in the South or warm weather states. The best explanation of it's use and meaning is to say good-bye and close a conversation. It is often used in the general collection of closing phrases such as, "OK, well, we'll talk later, take care, mmm-bye." As stated there is as yet no etymological history of this term.
The use of this extends as far North as North Dakota and has little to do with region. However, in the warm weather (generally Southern) states speech often becomes slower and more relaxed, along with much activity. [This explanation was given to me by several good Southerners.] Clearly the purpose of the mmm is to continue, in anticipation, the use of the B in Bye.
The /b/ being a bilabial stop or plosive consonant requires the lips to be closed to begin with. Rather than anticipate by saying "B-B-B-Bye" the laid back, relaxed method is to start the voice before the lips get into action. These leaves us with the calm "mmm" for as long as comfortable to be finished off with the plosive B, then Y and E, mmmmmmBye. A comfortable and relaxed conclusion.