The so-called V-speeds are calculated before takeoff; the 3 most prominent are V1 (the speed at which, during the takeoff roll, it is no longer safe to abort the takeoff, because you are going too fast. If anything goes wrong past this point, you continue the takeoff anyway), Vr (rotation, i.e. pulling the yoke/stick back, getting into the air), and V2 (the safe single-engine-failure climb-out speed).
Immediately before takeoff, the autothrottle is configured to maintain V2 speed; so no, an engine failure on takeoff is an event you are always already prepared for.
EDIT: For the 777 for example, and (that I know of) all other Boeing aircraft and probably Airbus too (but don't quote me on that), you can also see your V-speeds on the speed tape during the takeoff roll. This video shows them on the 777, they're the green numbers on the far left. The flight crew also calls them out. Finally, the green arrow pointing upwards or downwards is the projection of what your speed will be 10 seconds from now. Note how the programmed speed (top left, purple) is the V2 speed. https://www.youtube.com/watch?v=Uw9P-aMFYE0
Yes, that's depicted in the video I linked (THR REF is thrust mode). I didn't mean to imply that the autothrottle will only stay at V2 speed, but rather that the automation won't go below it.
But that's my point, automation doesn't hold V2 for you. On engine failure during takeoff you immediately disconnect the autopilot, then manually stabilize the yaw with rudder, adjust thrust and retract gear (and flaps in steps). Then manually fly the climb at V2 speed. All the automation does is indicate the correct speed on the PFD speedtape to help you decide what to do.
