Can my water softener's backwash cycle run while the well pump is out of service?
If it has a mechanical timer it doesn't need a UPS so I am figuring you mean a rotary electric timer as a opposed to a digital one.

As you have probably surmised the softener would go through its timing cycles even if the power is off but the well pump will be depleted somewhere during the softener's cycle and it will fail to properly regenerate. How far it gets through its cycle depends on how big of a pressure tank you have for the well. It is possible it won't even finish the brine cycle and you could have some brackish water until it clears out.