Now Im a fairly new boy to this, but this is what I imagine Id do:
All Id need is a "callsign, engine failure, standby". First thing would be the priority line to the radar controller and advise him of the engine failure. That way he can keep things clear of you. Id then vacate any lined up aircraft and clear any aircraft on short final to land. Anything further on approach I would 'go around' or ask radar to break off in order to clear the circuit for you. Next thing would be to know if you are going to fly a visual circuit or require vectors. If its a visual circuit, as soon as the circuit is clear Id clear you to land. If you need vectors Id see if you can accept a Freq change then put you to radar and tell them that you are cleared to land.
Well, thats the theory anyway...
At a free-flow airfield the climb out is pretty clear anyway so 99% of the time there will be no traffic (especially with reduced performance) and as soon as I let radar know they can move everything else out of your way. At the end of the day, you get the priority.