I've completed several missions that supposedly rewarded me with 4 engineers, but the engineer count (as shown in the Engineering display) hasn't increased. Am I doing something wrong?
No, what you are telling seems right. Engineers and scientists are awarded immediately after the mission. When you enter Engineering or Research after the mission, you can also hear the head of the division thanking you for new people.
If you are sure you choose the Engineers mission and that the number stays the same, it is a bug.
P.S.: There is an upsetting number of times I write "then this is a bug" in my answers :-(
Did you choose the mission using the arrow keys? The arrow keys cycle through the missions and their rewards, but only a click actually selects them. I missed out on my desired reward a couple of times before realizing what was happening.