I sure can't improve on you folks' advice, but once the plane is climbing with gear up, at least V2 and on runway heading, at least a few hundred feet in the air: the non-flying/handling pilot calls ATC and says: 1) "we're declaring an emergency, engine etc problem, request straight ahead on runway heading". They know that we'll call him/her right back when (cleaned up and climbing at Vzf, with the Captain ready to give the plane and the ATC radio to the FO, so the Captain can quickly read/do the Emer Checklist...) ready for a turn towards crosswind/downwind...and 2) "we need the emergency equipment standing by".
This is our typical simulator plan for an engine failure at or just after V1, or for an engine fire on takeoff. It will work if we lose the Emer AC or DC bus in IMC or whatever.
If tower/departure control sees an obstacle ahead (or the "Prohibited Airspace!" in Wash, DC), their JOB is to turn us NOW! If in uncontrolled airspace with terrain etc, you had better have your SID out and very VISIBLE on your yoke or Airbus "lunch tray"... anyway.
A former check airman I flew with, when I told him that it might not always be best to return a heavy airplane to the departure airport with only short runways (Midway or LaGuardia, NY only 7,000 feet) when much longer runways are at a nearby airport, definitely agreed with that idea, especially landing at hotter single-engine speeds etc, on wet or dry runways.
With changing/rushed duties, just make sure WHO is flying the airplane.