"The initial alignment is from the airport ARP, and is downloaded when you enter the airport four-letter ICAO identifier. "
Sounds a little odd, Oxford, but I'll take it on board. During maintenance, I've aligned the IRS's many times without entering anything in the FMC... other than the present position. If I forget to enter the position, I get a prompt in the scratchpad 10 minutes after switching on the IRS's (747-400). In this case, the full alignment takes only a few seconds after entering the position. Perhaps the IRS's take last known FMC position in lieu of the ARP?
What would happen in the case of a dual FMC failure on a 767? I can't see the alignment process being degraded or slowed down by this.... You'd simply enter the PPOS on the overhead panel (no ARP necessary for initial alignment).
Cheers.
Q,