To quote an answer by kevin on another question from last year:
What's the procedure? The procedure is, be creative to save as many lives as possible!
Really. The procedure is to determine a course of action which will likely result in the best outcome for everyone, utilizing all resources and given all constraints. Period. It is as simple as that.
There are infinitely many scenarios, and one cannot be trained for everything.
If you need to communicate something as quickly as possible in order to avoid a disaster, and calling "mayday, mayday, mayday" will be a significant waste of valuable time, then the answer is clear: don't call mayday. Just say what you need to say.
If you know what the other aircraft need to do in order to avoid a collision, perhaps it would be a good idea to announce "Go around, go around, go around!", "Cancel takeoff, cancel takeoff, cancel takeoff!", or "Hold short, hold short, hold short!", as appropriate.
In the case of Air Canada 759, where an arriving airliner mistakenly lined up to land on an occupied taxiway instead of the runway, a pilot on the taxiway announced: "Where's this guy going? He's on the taxiway." This probably wasn't the best possible thing to say, but the airliner went around and avoided a collision.