Have we underestimated the risk of a NATO-Russia nuclear war? Can we do anything about it?

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Until recently, I thought that the risk of a nuclear war in the 21st century, while not zero, was nevertheless very low and the marginal bit of effort spent reducing it further was probably not a good use of resources. But in the past two weeks, a series of articles on Vox.com have led me to rethink that view. The most detailed of them is worth reading for full context, but I think the key points are these:

  1. Some experts are starting to worry that recent events in eastern Europe have raised the risk of a NATO-Russia nuclear war.
  2. This is because Putin is feeling vulnerable and threatened, and is using nuclear saber-rattling to compensate, doing some things even Cold War-era Soviet leaders avoided (because they felt more secure in their position).
  3. While nobody wants an all-out nuclear war, a particularly worrisome scenario is that Putin does something which, he expects, will scare NATO into backing off, but instead leads to a spiral of escalation.

I've tried to see if other people in the EA community have thought about this issue. The main thing I can find is GiveWell's research into nuclear policy, but when they've looked into the risk of nuclear war, the focus has been on an India-Pakistan nuclear war (which admittedly could be pretty devastating, not just to India and Pakistan but to the rest of the world via fallout and environmental damage).

I'm not sure what to do about this, except to say that it might be nice if GiveWell looked into the issue, if only on the level of one of their "conversation with a single expert" reports. A longer-term strategy might to found an organization dedicated to shifting incentives towards politicians in the US, UK, and France towards less bellicose rhetoric and less escalation, and more international compromise. (One worry when looking at the risk of nuclear war is that politicians may be repeatedly tempted to do things that slightly raise international tensions, for the sake of scoring political points back home.) But it would take a special skill set to be able to found such an organization.

Does anyone else have any other ideas?