"Omm," the shaming word
Solution 1:
The comments thread has clearly informed you that there are parts of the US and the UK where the term is unknown. It isn't unknown to the internet however.
Bet they were the goody two shoes ‘omm I’m telling miss’ types when they were at school. singletrackworld.com
and
I was just humming this to myself when my niece came in and said 'Omm, I'm telling! You're singing that song that goes, 'I've been... amiright.com
and
Omm! I'm telling on you! Reddit
While I can't pull out any regional information from these, none of the occurences I found, and they were all in online conversations, seemed to cause any consternation or confusion amongst others in the threads.
While this does not fully answer the question, I am hoping it serves to help keep it open while we wait for an omm-sayer to happen by.
In the meantime I found a thread on thestudentroom forum where they were discussing playground insults etc
In the English Midlands people seem to say ‘amm!’ while others say ‘umm!’ And one person claimed ‘ommvv’ while another went with ‘omer!’ All with exclamation marks and definitely part and parcel of the ‘I’m telling’ rather than a preceding hesitation. Because people in the thread are writing in imitation of how they used to say it the spellings tend to be very drawn out; ‘uuuuummmmmmm!’ Which is leaving me struggling with how to even interpret this one
if someone swore at my school it was 'ammmmmmmmmm meeerrsssyyyy' or something alike XD and someone would tell
And finally, with apologies to @AndyT for not picking it up earlier, pretty much the first hit in his linked search is to a Playgroundjungle thread where people from across the Anglo-sphere chip in with their local variations, including someone from Scotland, though I swear I never heard or did this when growing up here.
My fiancee just used some of my dad's posh coffee, I said "Ummmma, I'm telling on you" and she didn't understand why I said ummmma (nor do I tbh). She's from South Africa so I guess the custom isn't used over there. I'm from UK, grew up in Scotland where I must have picked up the umma variation. Both grew up in 1980s
The earliest date given in that thread is the 1960s though, presumably due to the age of the contributors most claims are for the 1980s.
when I was a Primary School kid in southern England in the 1960s, the words “Ummmm! I’m telling on you!” were in common usage. And yes, the intonation used when saying “Ummmm!” was ‘rising then falling’.
Solution 2:
We used to say ommmomomomaw if someone was naughty! I’m from north wales and it’s still a thing here today in the primary schools, no idea where it came from but it wasn’t a thing in my previous English primary school so I had thought it was just a Welsh thing! Apparently not!