How does R handle NA in ifelse with 2 vectors?

Can someone please help me understand how R handles NA in ifelse statements?

Let's say I have the following vectors:

x <- c(1, 1, 3, NA, 4)
y <- c(1, NA, 1, NA, NA)

I want to create another vector based on these two vectors, and what I want the output to be is:

z <- c(1, 1, 2, 0, 0)

I have tried this:

z <- ifelse(x == 1, 1
            , ifelse(y == 1, 2, 0))

And I tried this (which seems really excessive to have to spell all of this out):

z <- ifelse(x == 1, 1
            , ifelse(!is.na(x)
                     , ifelse(y == 1, 2
                              , ifelse(is.na(y), 0, NA))
                     , 0))

In any case that didn't work either... Both of those return:

> z
[1]  1  1  2 NA NA

Can someone please help me understand not only how to get the result I want but also how R treats NA when working with ifelse involving 2 vectors?

Thank you.