After the battery drops out, apart from loss of standby instruments etc does the FADEC and hence engine control drop out as well?
Nope.. runs on indefinitely.. or, rather, until the engines run out of fuel...
As long as the engine core rpms (N2) remain above 9% (CF6) or N3 remains above 8% (RB211), the ECU/EEC's are powered (courtesy of engine-driven permanent magnet alternators).
But if the rpms go this low, you have probably fallen out the sky already... and you can probably assume that the engine has flamed out.. and without battery power, you won't be able to relight.
Rgds.
NSEU