NATO is not going away, There one major question, and a few smaller ones. The major one is: Where will post-Trump USA go?
Troubles around the Baltic, Black, and Mediterranean seas are naturally going to be felt more closely by Europe than by North America. US interests will more likely go to East, South-East, and South Asia, and belatedly Africa.
The Monroe Doctrine seems forgotten, at least for the moment, but it might return in some shape. (Side note: South American trade with EU is larger than with the US. Take that and smoke it, James!)
While the perspective and priorities differ, US and EU are much more aligned than you would expect two blocks to be. And in the fuzzy things, values, they are even closer, and they are converging, not diverging.
https://www.youtube.com/watch?v=AiIpymGeGooBrexit and NATO is another question. By the look of it the British are making an absolute mess of things, and will need someone to blame. That will pass. But without NATO, veering between EU and US would be trickier. NATO is a good framework for heterogeneous members (Turkey after all is a member too).
There will have to be some accord with Russia at some point. Not likely soon though.
Turkey is tricky, even after Erdogan, with ups and downs (hopefully more of the former).
If there were a US/EU divorce, NATO would still exist as a (likely French-dominated) European army.