Apple TV behind wired router not available for wireless connect devices and iTunes

A picture of the network setup would help. Also post the IP address of the Apple TV, the IP address of one or two wireless devices, and the IP subnet masks (look in Network settings).

it's likely that your router keep wifi and wired ethernet on separate networks and you need to go into it's settings and bridge them

when you do they should all be on 192.168.178.X or 192.168.71.X

it's 100% a network thing and can't be fixed from the connected devices

For it all to work everything needs to be on the same subnet, either 192.168.71.* or 192.168.178.*.

Without some sort of illustration I can't tell for sure, but it looks like you have both routers in "router" mode. What most people do in their home setups when two such devices is this:

- The router closest to the cable modem or whatever equipment you have from your provider stays configured in router mode.

- The second router gets configured in "bridge" mode so that everything connected to it is on the same subnet as the router.

Then everything gets assigned an IP address in the same range and all of the devices can see each other.

It looks like the one you call your "wired" router is the one that should be in bridge mode. If you can't figure out how to do that, let us know the model of router and someone here should be able to point you in the right direction.

The Apple TV relies on the Bonjour service (also used for things like networked printer services) which only covers the broadcast domain of the network on which it resides. So only devices on that same subnet get the advertisements that it is there, and therefore they are the only ones that can connect.

Is there a pressing reason to have the ATV on the network with the NAS?