One of two things can happen.
1) Aircraft accelerates quickly, and maintains 1g, essentially maintaining its previous trajectory.
2) Aircraft maintains V.a.prot, but due to the increased thrust, and the fact that I have not disconnected the protection (by using forward stick), aircraft pitches up into a steep climb, still maintaining V.a.prot.
#2 is closer, but it is the AoA that is maintained not the airspeed.
V.a.prot, the steady-state airspeed corresponding to a.prot, is only calculated for display on the airspeed tape, it is not the control target.
In a non-steady-state condition the actual airspeed can be quite different.
See
Air Accidents Investigation: Airbus A330 C-GGWD and Airbus A340 TC-JDN
for an example where turbulence caused an A340 to exceed a.prot during cruise. Alpha prot engaged and the aircraft zoom climbed over 2000 feet before the PF disengaged the mode, with airspeed well above v.a.prot the whole time.