Quoting Hyperveloce:-
the maintained NU inputs are the PF inputs
Not your fault, Hyperveloce, because the BEA Note leaves so much out. But the only mention of a 'left noseup input' at any stage before the upset is this:-
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 next relevant entry, referring to at least 11 seconds later, says:-
At 2 h 10 min 16, the PNF said "so, we’ve lost the speeds" then "alternate law […]".
Had the PF maintained a 'left noseup input' throughout those eleven seconds the aeroplane would probably have been 'standing on a wing.' But the PNF appears to have been (relatively calmly) going on doing his job and reading and passing on the messages appearing on his screen. That suggests to me that the PF had in fact merely corrected a tendency to roll after the autopilot signed off, and then levelled off and followed the prescribed procedure of 'flying pitch and power.'
The next entry in the BEA note (with no time stated) says:-
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.
No mention of any 'nose-up inputs' - indeed, that entry strongly supports the view that the climb was 'uncommanded' and that the PF took appropriate action to counter it?
The PF is reported, much later on, as maintaining 'noseup inputs,' but that was after the stall had occurred. And, as I said earlier, I think that the most likely reason for those is that the pilots thought for quite a while that they were in a dive rather than a stall.
So IMO, as BOAC said, "The climb is still unexplained."