Here's my theory then:
(bearing in mind I have no knowledge of the 777 fuel system - but that won't stop the D*ily M*il from quoting it):
Some seals/joints were deformed during the extreme low temperatures earlier on in the flight. The seals however had lost flexibility and still maintaining a seal, or at least weeping only slightly. As the atmospheric pressure was low at altitude, the weeping would be from the fuel system to the outside.
As the aeroplane descended, the seals warmed up, regaining flexibility then flicked back, but this caused an inleakage of air into the fuel lines. The inleakage of air caused the pump cavitation observed.
By this point also the water that would have been thawing in the tank at this point - perhaps by putting the landing gear down it caused more condensation to drop from the walls of the fuel tanks.
Anyhow, going back to the air. As the air was drawn into the inlet side of the high pressure fuel pump, this induced sudden foaming exacerbated by the presence of the water in the fuel lines. This choked the fuel pumps and disrupted fuel supply.
Tiggermoth
If this had been the case, then the way to have recovered from the situation would have been to stop demand on the fuel system, allowed the fuel to fill the suction line, then slowly put an increased demand on the pumps while throttling back the discharge side. As the autopilot just put a greater demand on the fuel system, it just made it worse.