It only works for single deliveries that have multiple letters, e.g. some of the Irongate Blacksmith deliveries. It won't let them do two different single-letter deliveries at once.
If you're selecting a letter from a multi-letter delivery request, and your bird can carry multiple letters, it will. I haven't seen anyone else post a bug about this and it's working fine for me. So this is a PEBKAC issue, afaict. You wouldn't have to wait, it happens the instant you send the bird off with a letter from a multi-letter delivery request. The bird doesn't fly down to pick up the other one, it just goes with them without any visuals other than the second one disappearing along with the first.