[r] How to include NA in ifelse?

You can't really compare NA with another value, so using == would not work. Consider the following:

NA == NA
# [1] NA

You can just change your comparison from == to %in%:

ifelse(is.na(test$time) | test$type %in% "A", NA, "1")
# [1] NA  "1" NA  "1"

Regarding your other question,

I could get this to work with my existing code if I could somehow change the result of is.na(test$type) to return FALSE instead of TRUE, but I'm not sure how to do that.

just use ! to negate the results:

!is.na(test$time)
# [1]  TRUE  TRUE FALSE  TRUE