Once a ward ability has triggered, it doesn't matter if the artifact loses ward (most likely because Bronze Guardian leaves the battlefield in response) before the ability resolves. The controller of the spell or ability will still have to pay or the spell or ability will be countered.
2021-04-16
The ability that defines Bronze Guardian's power applies in all zones, not only while it is on the battlefield.
Double strike
Ward (Whenever this creature becomes the target of a spell or ability an opponent controls, counter it unless that player pays .)
Other artifacts you control have ward .
Bronze Guardian's power is equal to the number of artifacts you control.
or the spell or ability will be countered.