From what the OP describes I would think the flightdeck crew were busy trying to fight the unusual attitudes due to the turbulence for the first minute or so of the incident. After all they are also just human and the seatbelt sign might have slipped their mind(s). If Clear Air Turb hits when cruising near maximum performance altitude there isn't much "room" to stay out of both high and low speed buffet zones on the speed tape/airspeed indicator and that would have held their attention for sure!
The fact that engine noise died out could just be due to throttling back if an overspeed was imminent (often the case on my 737NG when caught by a sudden wind change that can accompany turbulence).
Terryp - why don't you post in SLF or questions? - maybe including the flight number and date in the title might get an answer from a BA person with more info.
I do think it's a bit of the poor show if you heard nothing from the flightdeck crew....