How about the old golden rule of aviate, navigate, communicate?
Why are people so fixated on calling a mayday, when flying the plane and fixing the problem is more important then telling ATC initially you have a problem.
In RYR we use PIOSEE, NITS, PA - which I think is pretty good at keeping it all coordinated.
P Problem
I Information
O Options
S Select (an option)
E Execute
E Evaluate
Once this is done, tell cabin crew using the NITS brief
N Nature of emergency
I Intentions
T Time available
S Special instructions (SOS/ditching demo etc)
Then PA to pax.