In the UK, the term "Oh my God!" is used extensively, and is often abbreviated, particularly by the younger generation, to "OMG" in all forms of their e-communication. It is used to express a very wide range of emotions: from extreme pleasure/shock/disgust/fear, all the way to mild surprise or anticipation. The user will not, generally, have any consciousness that he/she is appealing to, or referring to, a deity.
The expression "Oh Jesus!" is not nearly so commonly used in the UK - perhaps because the religious aspect is more specific and apparent. When it is used, it is most likely to express extreme shock, disgust, fear or sorrow. The expression "Jesus Christ!" is more common in these extreme situations.
As an aside, my father (a staunch Scottish Presbyterian) swore rarely and never blasphemed - apart from the very few occasions when extreme frustration caused him to utter "Christ on a bicycle!" I have never come across this expression elsewhere....
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