Not sure about his wireless router, but none of the four I've had have an uplink port. Neither did the non-wireless capable one I had either. The newer router's ports are auto switching though. (The old ones were not, so I had to use a crossover cable.)
I've done what you're asking about. I just ignored the WAN port on the wireless router and connected a LAN port on the (original) non-wireless router to a LAN port on the (new) wireless router. I shut off the DHCP server on the wireless router (you don't want two DHCP servers) and gave the wireless router's LAN port a fixed IP address that the wired router's DHCP server doesn't serve so there would be no conflict. The other PCs hooked into the wireless LAN's switch and the wireless PCs (laptops in my case) were able to broadcast for and get a DHCP-assigned IP addresses from the DHCP server on the old wired router.
I can't recall if there was a way to or I had to set the gateway address (and DNS servers). You shouldn't have to. I think I set that all up in the fields for the WAN (Internet) side of the wireless router just in case. I set its IP address the same as the LAN IP address. (There was never a cable connected to the WAN port, so setting that really shouldn't matter.)