If uncommanded by PF (or a/c), the climb may have been addressed with PF's 'Nose Down inputs'. I think BEA have not specifically timed those.
Bear, to be clear, that depends on the duration and frequency of nose down inputs as counters to nose up inputs (per how a Sidestick works) which detail is still vague. The BEA seems confident that at least initial climb was initiated by a nose up movement of SS.
The second STALL WARN, could actually be predicting the STALL that resulted from loss of energy in this climb (which thus far is not described as to origination). The AoA (not necessarily PITCH) would increase as the Plane slowed, independent of elevators that had lost their effectiveness anyway.
Huh?
The climb slowed the plane. Since power was not initially reduced, climb came from pitch up, so ... AoA change was due to ... drumroll ... pitch up. You with me? It has been suggested that when TOGA was selected later in the event chain, nose pitched up more due to how planes like this fly. This tends to increase AoA if not countered by the appropriate pitch adjustment (n-d) to accompany the power increase. This is Flying 101, and would be called for if in alternate versus normal law. There is evidence that pilots were aware of being in alternate law. Not sure of the granularity with which BEA can parse the data from FDR ... we shall see. I suspect it correlates well enough.