The person credited with creating the term was a mathematician, (and also extraordinarily hawkish). He was almost certainly aware of Nash's work, as he coauthored a book proving the existence of Nash equilibria for zero-sum games prior to Nash's more general proof:

https://en.wikipedia.org/wiki/John_von_Neumann#Mutual_assure...

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As a note, I can't find a date for Von Neumann coining the MAD term. It seems likely to me that it would predate Nash's work, though as I said earlier, the existence of such equilibria was already well known (It was jut not proven that for large classes of non-zero-sum games there must exist at least one such equilibrium, which was Nash's contribution).

I have never seen the phrasing "mutually assured destruction" in game theory (outside of nuclear references). However, outside of edge cases, it is the same concept as a Nash equilibrium. Specifically, it is a strategy where defecting leads to a worse outcome for both parties.