Divergence in spelling would probably just have followed the divergence in pronunciation. To add more to the mix, in Ireland I've seen it as "mam". In the UK we tend to use "mum" / "mummy".
The reason I raised the question was that the U.S. early settlers were mostly from U.K. and they would have the same dialects and pronunciations as those in the U.K. I was curious as to how and when the divergence took place.
A quick Google of the etymology (etymonline.com, for example) suggests "mum" was first used as a shorthand for "mother" in 1823. By then the ongoing difference in pronunciations of words from one side of the pond to the other was presumably already well established. "Mom" seems to have been first recorded in 1867. It doesn't say in which country these occurred, though.
