This is an interesting scenario, and one that has not been unexplored in fiction. However, you ask, "How would society react if the god of one religion has been scientifically proven?" but then immediately introduce the assumptions that "the scientifically proven god is omnipotent and omniscient" and "the god is a singular, immortal being." For some religions this simply isn't the case - not only with fictional religions, but with actual religions as well. Many early historical religions where polytheistic and didn't attribute things like omniscience or omnipotence to any particular god. For a modern instance, in the Mormon religion, there is not a single God but a hierarchy of Gods, of which only one is "ours".
The various Star Trek TV series had more than one separate instance of human encounters with such deity. For instance, Star Trek:The Next Generation introduced the apparently omnipotent and omniscient "Q". He was neither worshipped nor treated with reverence by any significant number of people who encountered him on the show. However, you ask specifically about a whole society.
This is explored in Star Trek:Deep Space Nine, where the whole Bajoran society is based on the "prophets" who are essentially gods who live in an artifical wormhole connecting the Alpha and Gamma quadrants. There is proof that these gods exists and live in the wormhole, and their power is manifest from time to time. Despite this proof there are still non-believers (outside of Bajoran society). Are they gods or are they worm-hole aliens? Those who don't believe say the latter. 752
The "emissary" says, "Why can't they be both?"
There is a particularly interesting dynamic in ST:DS9 because later in the series we are also introduced to the "Founders" - of non-solid shape-shifters who have genetically engineered a race called the "Vorta" to administer their Dominion, of which the most interesting is the "<PERSON>" character. Here are a couple key quotes:
<PERSON> [a shapeshifter]: Has it ever occurred to you that the reason you
believe the Founders are gods is because that's what they want you to
believe? That they built it into your genetic code?
<PERSON> 6: Of course they did. That's what gods do. After all, why be a god
if there's no one to worship you?
<PERSON>: Pah-wraiths and Prophets. All this talk of gods strikes me as
nothing more than superstitious nonsense.
<PERSON>: You believe that the Founders are gods, don't you?
<PERSON>: That's different.
<PERSON>: [laughs] In what way?
<PERSON>: The Founders *are* gods.
To answer your specific questions.
How would believers of other religions react to this? Rivalry perhaps, or dismissal of the evidence.
How about atheists? Interestingly enough, I don't recall there being any Bajoran athiests, and that society had had actual documented interactions with their gods over time, including supernatural "orbs". So from this standpoint, it's likely that if there were not any cultural differences, people would convert. From a scientific standpoint, the thing that prevents a lot of people from not being atheists is the lack of evidence. If you provided irrefutable proof to <PERSON> that God exists, he would have to believe. That doesn't mean that he might not have issues with God, but I don't think he would any longer claim he doesn't exist.
Would monotheistic religions be abolished if the proven god isn't theirs? Again, this is probably a cultural thing. If a different culture's diety was proven true, this doesn't necessarily provide any incentive for a culture to abandon their own, unproven diety.
How would society at large react to this? How about the law and government? This probably depends on the nature of the proven deity. For instance, if the God of the Christian Bible was proven correct, then all the other things written in the Bible about how law and government would work could be expected. 72