R - programmatically detect NA columns and return string

I have this vector of eligible columns for my script

cols <- c("country", "phone", "car")

And this dataframe

test <-
  data.frame(
    id = c(1, 2, 3),
    country = c("us", NA, "uk"),
    phone = c(1, 1, NA),
    car = c(NA, 0, 1)
  )

Goal is to create new column with result, where condition will be based only on columns present in cols variable. In case that all values for id are NA, then res should be string nothing, if some of them are not NA, then I need to this colnames, in case that all columns are not NA then result should be string all.

result <-
  data.frame(
    id = c(1, 2, 3),
    country = c("us", NA, NA),
    phone = c(1, 1, NA),
    car = c(NA, NA, NA),
    res = c("country, phone", "phone", "nothing")
  )

I can do it only via case_when() function

mutate(
    res = case_when(
      !is.na(country) & is.na(phone) & is.na(car)  ~ "country",
      T ~ "?"
    )