Bono:
Suppose no one was alive on board, but it was happily cruising along at 29500 ft, heading south.
The autopilot would keep this going as long as possible. When fuel starvation sets in, one main tank will be empty before the other main tank. This is because one engine will always consume more, fuel flow wise, than the other.
The first engine flaming out will be corrected by the TAC. No problem. Also, a 777 at zero fuel weight (ZFW) will be able to maintain FL295 with one engine.
A few minutes later the other engine flames out. All generators will drop off, so no electric power is provided. However, the RAT will kick in, providing electrical power to, also, the autopilot. (Actually, only the transfer busses, but the Autopilot is on this)
The autopilot will try to maintain FL295, but is unable to do so due to lack of engine thrust. When top of yellow band is reached, the autopilot will slowly descend with this speed until impact. This is also what happened in london with the BA 777 on the ILS when it flamed out. The A/P flew the aircraft against the ground at heathrow with minimum speed.
All a bit rusty, I last flew the 777 three years back.