Do stock markets price in existential risk (i.e. global nuclear war)?
I don't have anything to say on the empirical side.

On the theoretical side, I think that is an interesting question, even in a standard framework with "rational" agents. The possibility of the end of the world reduces the expected stream of income from assets (e.g. a share of a firm) because maybe the firm will not be there any more to distribute dividends.

Maybe you will not be there (that's the existential risk). About this the prediction of the standard microeconomic theory is the same as in the science fiction movies : your preference for present should rise, the 'natural' interest rate and the actualization rate with it. (If you fear to be dead tomorrow, you will not lend your money today... unless for a very high interest rate.) That means the value of future income falls and you want to consume everything now.

So both effects go in the same direction and "standard" prediction would be that the stock prices should fall.