Bearfoil
From the outset and per BEA the PF made one NURL at a/p drop. It was not continuous. The climb is unexplained, although High speed Prot is a candidate, certainly.
The report does not explicitly say how long the nose-up input was. IMHO it was continuous from 2h 10 min 05 "...nose up input." until 2 h 10 min 16 "...nose-down control inputs". At most this was for 11 seconds and resulted in the climb from FL350 to FL375 which sounds reasonable.
We can speculate if/why the PF intentionally made the climb as he did not adjust the thrust during the change in FL despite announcing he had control before the manouver. This initial nose up would appear to be counter intuitive given the stall warning and desire to maintain cruise altitude. Unfortunately the IAS on the right PFD used by PF was not recorded on the FDR. Did the ADIRUs give bogus attitude indications? It would explain the bizarre "inputs made by the PF were mainly nose-up".
From 2 h 10 min 05 , the autopilot then auto-thrust disengaged and the PF said "I have the
controls". The airplane began to roll to the right and the PF made a left nose-up input. The stall
warning sounded twice in a row. The recorded parameters show a sharp fall from about 275 kt
to 60 kt in the speed displayed on the left primary flight display (PFD), then a few moments
later in the speed displayed on the integrated standby instrument system (ISIS).
Note 1: Only the speeds displayed on the left PFD and the ISIS are recorded on the FDR; the speed
displayed on the right side is not recorded.
Note 2: Autopilot and auto-thrust remained disengaged for the rest of the flight.
At 2 h 10 min 16, the PNF said "so, we’ve lost the speeds" then "alternate law […]".
Note 1: The angle of attack is the angle between the airflow and longitudinal axis of the airplane.
This information is not presented to pilots.
Note 2 : In alternate or direct law, the angle-of-attack protections are no longer available but a
stall warning is triggered when the greatest of the valid angle-of-attack values exceeds a certain
threshold.
The airplane’s pitch attitude increased progressively beyond 10 degrees and the plane started
to climb. The PF made nose-down control inputs and alternately left and right roll inputs. The
vertical speed, which had reached 7,000 ft/min, dropped to 700 ft/min and the roll varied
between 12 degrees right and 10 degrees left. The speed displayed on the left side increased
sharply to 215 kt (Mach 0.68). The airplane was then at an altitude of about 37,500 ft and the
recorded angle of attack was around 4 degrees.