Angels and demons affect reality whether they're trying to do so or not. That's why the Bentley was sentient and had the perfect Queen hit for every moment. It's why Whickber Street fell apart after Aziraphale left.
In those last few seconds, Aziraphale and Crowley were saying fond goodbyes. They were thinking of each other -- of the depth of their history, and of how very together they were, and, a little bit, of their regrets.
The other side of a regret is a wish. Specifically, it's a wish that things had been different. And when angels and demons wish for things, reality often has a way of bending around them, even if all they're thinking of is each other.
A few years ago, Aziraphale and Crowley desperately tried to hide a little half-miracle from Heaven; they weren't completely sure of themselves, and they had the awkward distraction of dealing with Jim in the meantime. That teeny-tiny little half-a-miracle ended up warping reality the same amount as if they'd brought Lazarus back to life 25 times.
This time, Aziraphale and Crowley weren't hiding anything; they were united and committed in front of their very maker, and their thoughts were pure. They loved each other, and they loved humanity, and they wished they'd been allowed to choose each other like humans do.
Reality listened. In effect, they held hands with no one between them and did an 8 billion Lazarii miracle. They brought themselves and the universe they knew forward into the universe being created anew.
It took a while (13.8 billion years), but the stuff that had once made up people and histories who had disintegrated before our eyes eventually reintegrated. For all the power of the Book of Life, Aziraphale and Crowley weren't undone. The world had a connection to Aziraphale and Crowley, and Aziraphale and Crowley made a connection to the next universe. And like Michael carrying the Archangels with her, they carried the world with them. It wasn't a one-way relationship, either; the remaining hints of the wills of people who had been erased clung to that one little bridge for dear existence. That's what free will looks like when there's nothing else left.
It wasn't a perfect duplicate of the old universe. The people couldn't be angels or demons, of course, so the angels and demons all got stirred into humanity. But it turns out a sapient being is a sapient being. Everyone was also starting from scratch, experience-wise, which leads to questions about whether they were the same people at all. But they were certainly continuations of the same stories.
You could argue that this means there was technically celestial interference in humanity’s next course, but all the quantum-level stuff that was brought over from the previous universe had to slog through billions of years of the laws of physics before any of it came together in a human way we would recognize, and it could have been stopped at any time if it weren't made of the echoes of people trying really hard, so I think it's earned its place in this universe, too.
As for God, She's on the outside, off doing something ineffable somewhere else. She knew right away that the new universe was made of the same stuff, including the same people (for starters, anyway) as the previous one. But She had already set the parameters: She could not interfere.
Sometimes She peers in to see how the slow-cooking universe is doing, but She doesn't have any particular plan. Sometimes, it's more interesting to see what happens when you don't stick your nose where it doesn't belong. It turns out that She doesn't have to mess people around for their lives to be worth witnessing.
