Have heard this said a lot on TV lately, first heard it on an advert about a year or so ago but didn't recognise it as an actual saying. Where did it come from?
I can't find any reference to it in the OED but Wiktionary cites the earliest reference as "Oh, my days, don’t ’ee say another word about either of ’em. I wish I’d never been born" in The Wingless Victory, by Mary Patricia Willcocks, published in 1907. That suggests that it's a northern phrase (of uncertain origin) that has recently found wider usage through it's use by certain people with celebrity status.
It's a phrase I use from time to time. My old boss was a Church Elder and quite proper - a good laugh and very decent man, so I changed my "Oh my God" to Oh my days so that I wouldn't offend him.
My great granny used it all the time. That and "suffering cats". No idea where she got that one from.
My take on it is the same as heathfields, as it's usually said about something that is hard to believe, in my experience anyway. Having said that there are often regional variations of uses, and I have heard it used as an alternative to swearing or blaspheming as jno said.
I heard somewhere that TV stars & contestants were told to avoid saying 'Oh My God' in case it caused offence to viewers, so mayb its one of the alternatives they have chosen to use.