It’s a bit embarrassing, but that already did the trick. I could indeed not see any USB devices on the remote system (=server). After waking up from suspend mode everything worked now, i.e. I could see the devices as well as connect the SpaceMouse.
Just for reference purposes: For some reason I needed to install the official 3Dconnexion drivers from their webpage to make it work via NoMachine even though it had worked without it when I connected the mouse directly to the server PC before.