think of it like this:
Backup is a
modifier
, it modifies the attached role to be only usable when a role of that type exists. If a 1-shot Loyal Doctor dies, a Nurse (special name for a Backup Doctor) gains a now ungated and non-Loyal Doctor ability. A 1-shot Nurse has a 1-shot ability even if the Doctor already used a shot.
Universal Backup is a
passive role
, it can receive modifiers that make sense for passive roles but it will inherit the modifiers (and used shots!) from the role that died. If a 1-shot Doctor that is depleted dies, the UB will be depleted. I'd assume an Even Night UB that inherits an ungated Cop can start cop checking on even and odd nights.
A UB related question: Is there a non-random way to resolve a UB inheritance when multiple players that a UB can inherit from die during the Night?
Extreme minutia: Would a Backup UB become able to inherit a role when a UB inherits a role?