This set up should work; in fact, I've done similar things myself.
I suppose my first question, though, is whether the wireless router has any wired ports. Typically, wireless routers will have four or eight ports for wired connections in addition to the wireless connection. He might be able to dispense with the first wired router.
Typically, if you switch routers connected to the modem, the modem has to be rebooted/power cycled.
If he has to stick with his current setup, then I several avenues of investigation.
The first thing I would look at is the IP addresses assigned to each interface. I'm guessing that the wireless router is confused because it has the same subnet (ip address range) on both the outside and inside interfaces. Most of these home routers come with 192.168.1.x preconfigured, and if the linksys offers 192.168.1.x to the netgear's outside interface, and the netgear is already using 192.168.1.x for the inside interface, it won't work. Change the inside (private) range on the netgear to something like 192.168.2.x.
Other possibilities include DNS not working, but I really think the most likely problem is an IP address conflict.