Wired Networks would use a router with a built-in switch, or wired through a switch.
A switch moves traffic only between the source and destinations without re-broadcasting it to all others machines wired up. This is in contrast to an older hub, which rebroadcasts every packet to every port. Hubs went out with 10BaseT; 100MB Ethernet and faster rarely uses anything other than switches.
On a wireless network, like on a ‘hubbed’ network, only one machine can broadcast at a time, and everybody has to wait for their “turn” to talk. On a wired and switched network multiple concurrent conversations can happily exist as long as they are all talking to different machines (ie different sources and destinations).
So, no, you would not get these problems on a wired network when moving files between computer A and computer C whilst playing content on your WD from computer B.