All right, this is going to be a cliff-notes version with no citations to the lore, and I'm prolly going to get some stuff wrong because I'm rushing it, but here we go.
We're going to go mostly chronologically on this I suppose, so from the beginning, the world of Fallout was very similar to our own until nuclear apocalypse.
The bombs fell, radiation and FEV mutating the various lifeforms of the earth into those encountered in the game. There was likely a fair amount of geographical changes incurred by this as well, the residual heat from the bombs likely changing sea levels, and potential geothermal activity being triggered at the sites where the majority of the munitions fell.
The real change, however, came several hundred thousand years after the war. The radiation has cleared up, the mutations have mostly stabilized, the ancestor species of those in the Elder Scrolls have been established (DeathClaws evolved into dragons, argonians, and akiviri snakemen, Supermutants have become the ancestors of giants, trolls, ogres, orcs, Men have split into humans and elves, etc.)
Wham, here comes the second moon. Advanced beings from far away have arrived within our solar system; potentially traveling on the moon, or just bringing it with them for shits and giggles. the introduction of a new heavenly body seriously fucks with the tidal forces and the continents. Eurasia becomes Tamriel after northern Africa combines with Europe, the islands of southeast asia combining into the swampy southern coast of the massive continent. Southhern Africa breaks free and becomes Yokuda, the Americas (complete with the return of the Cretaceous NA inland sea) become Akavir, australia as Lyg, etc.
The advanced beings fuck around in our solar system for a bit, establishing their own "planes of Oblivion" on the various planets and moons, harvesting and genetically engineering the denizens of earth. A few thousand ears of theat, and you get into the recorded history of the Elder Scrolls.