Google is being difficult.
In general? None share a cd. In M+ or raid? All of them share a cd. The CD in that content is by group not by spell.
If you’re asking who can bres, Druid, lock, DK, and Paladin
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Battle rez has a shared, group-wide stacking charge in instanced combat. So anybody who can brez can consume one of the charges, but then it’s consumed for the group.
Worth noting that a shaman self-rez doesn’t consume a charge, which is something I someone learned a week ago 
I think it would be nice QOL that “patch-up” would also refresh battleres charges.
I agree, since you’re past the timer at that point. If they’re going to give you Hero again, they might as well give you brez. At the very least, ensure you have a minimum of one charge.
Neither does Ancestral Protection totem (if you die in totem’s radius you get rezzed) or Holy Priest’s Afterlife (you can rez an ally while in Spirit of Redemption) or Restitution (you come back to life after Spirit of Redemption)
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What about the Ultimate Gnomish Army Knife? I think that can brez still?
The army knife can’t be used in combat, but engineering combat rezzes do count against your charges.
You earn a charge every 10m you’re "in combat. This is straightforward in raids, but mythic plus is considered 1 encounter and dropping combat changes nothing.