As is well known and as was stated in a few answers on this site (see e.g. here), Tom Bombadil was unconcerned about the War of the Rings and whether or not Sauron was defeated.
Tolkien himself put it this way:
I might put it this way. The story is cast in terms of a good side, and a bad side, beauty against ruthless ugliness, tyranny against kingship, moderated freedom with consent against compulsion that has long lost any object save mere power, and so on; but both sides in some degree, conservative or destructive, want a measure of control. but if you have, as it were taken 'a vow of poverty', renounced control, and take your delight in things for themselves without reference to yourself, watching, observing, and to some extent knowing, then the question of the rights and wrongs of power and control might become utterly meaningless to you, and the means of power quite valueless. It is a natural pacifist view, which always arises in the mind when there is a war.
The Letters of J.R.R. Tolkien 144
But do we know what would have actually happened to him if Sauron ended up ruling Middle-earth? It seems not unlikely to me that Sauron would have feared him and would have tried to have him killed.
Did Tolkien ever mention what would have happened to Tom in such circumstances? Is there a way to estimate this based on things he had written?