Abnormal law:
While the fact that Abnormal Law and the dropping of autotrim, *seems* like a possible explanation for the THS remaining +13NU, there is no mention of an law change beyond:
At 2 h 10 min 16, the PNF said "so, we’ve lost the speeds" then "alternate law […]".
While we all recognize the BEA note's undeniable brevity, I also believe that everything included and, perhaps just as importantly, NOT included has relevance - therefore my position, until TOLD otherwise, is the aircraft was in Alternate Law and remained so until impact.
THS trim:
I don't have all the control rate information in front of me, but the premise of the autotrim THS function is to ensure pilot elevator authority over the full range of travel. My understanding is this: If I demand NU, the elevator starts moving, keeps moving and eventually will approach and then achieve the demand, at some point a little later the THS starts to off-load elevator travel. The assumption being that if I continue to demand NU, then I probably want the aircraft to trim that way.
So let's say I now throw in some ND stick. The elevator will move ND and, just as NU will eventually achieve the demand. If left ND, the THS will again start to unload the elevator by moving ND, but clearly there is a time hysteresis involved. So a quick ND input will NOT cause the THS to move. A sustained input will. Otherwise we'd have the THS chasing the elevator all day.
Hence, to those questioning why the THS remained at +13NU, there simply was never a long enough ND input to cause the necessary demand to move it. Any initial demand was handled directly via the elevator.