When do artifacts lose hexproof when Padeem, Consul of Innovation dies?
tl;dr: Padeem dies from lethal damage before triggered abilities do anything, so the artifacts will not be protected by Hexproof.
When combat damage is assigned, Brood Sliver's ability triggers. However:

603.2. Whenever a game event or game state matches a triggered ability’s trigger event, that ability automatically triggers. The ability doesn’t do anything at this point.
603.3. Once an ability has triggered, its controller puts it on the stack [...] the next time a player would receive priority.

State-based actions, such as Padeem dying from lethal damage, occur right before any player gets priority (see 704.3). Thus Padeem is off the board, and its static effect is gone, before Brood Sliver's ability is even put onto the stack, and long before it actually resolves (which is when the artifact-destruction ability would trigger).