Obviously,no need to start to relight if you ran out of fuel
I would think it is obvious.
We are going to so many different alternatives here it starts to feel like standard
Chinese opc.
Now,ash entry or any other failures must be dealt with priority that hopefully all captains can deal with.
In the case of ASH and subsequent single or total engine loss,first priority is obviously to get out of the ash...follow QRH,then relight initiation.
You cant relight still in ash and you cant relight without oxygen in your system.
Same if both engines fail,both gens will go offline,cabin will leak,having an eye on the leak of the cabin will allow to judge if your cabin will reach 10000 within your NNC and therefore if donning the madks earlier is a sensible option or not(use of emerg or 100%).
With loss of both gens,capt will be (become)pilot flying and i hope it is so in all your manuals/sops...