"Engine Failure!"
I flew with a UK airline out of LHR for many years (B737 and A320). The EFTO (Engine Failure on TakeOff) was practised several times on every 6-monthly sim check. The engine failure was naturally, usually at V1 or very shortly thereafter.
The training was that NH would call out "Engine Failure!" and nothing more. The crucial decision of 'which' engine was left until later. Meanwhile, the PF would simply (!) fly the plane until a safe point when both pilots could analyse the problem and carefully decide which engine had failed. Then the shutdown procedure would take place, monitored by both pilots. This system hugely reduced the chance of an early and incorrect decision on which engine had failed, but both pilots were 'in the loop', being aware of the problem.
It was literally a "sit-on-your-hands-and-fly-the-plane" scenario until it was safe to analyse the problem. The old adage "Fly – navigate – and communicate" was always the priority in our airline.