Cops can only operate within their own borders (like a NY cop cannot enforce the law in Canada) unless they’re granted special permission from Ottawa to the White House on catching high profile fugitives, even then: their powers are limited in order not to violate their sovereignty. Would their badge hold any weight while they are overseas (who have their own judiciary)?
That’s only speaking on official terms: but, what happens if an American on vacation (like off duty sheriff) witnesses or encounters criminal activity while they’re abroad? Are they allowed to detain that person or do they have to eventually turn them over to local authorities? For instance, they could flex a LAPD badge in Japan, but is that valid or considered null?
In the UK anybody can make a citizen’s arrest if it isn’t reasonably practicable for a constable to do so.
However it can only be for a crime that is indictable (can be tried at a crown court). So things like significant theft, burglary and assault.
A US cop has no additional rights to enforce the law than anyone else.