Redundant network handling in linux box

You need to put two things : activating the forwarding (in /etc/sysctl.conf net.ipv4.ip_forward=1) and activate the NAT with iptables (/sbin/iptables -t nat -A POSTROUTING -s 192.168.1.0/24 -o eth0 -j MASQUERADE). The NAT translate the private network to the external IP, so the internal machines can go on Internet. This line should be applied by your firewall. I don't know which firewall you have, so you must integrate it in your configuration.

Edit: Add OP's solution from comments:

• Remove eth0 (wired) from NM
  • NM_CONTROLLED=no
• Left eth1 (wireless) controlled.
  • The default when NM_CONTROLLED is omitted
• Set eth1 METRIC=20
• Produces 2 default routes where eth1 takes precedence

Answer:

Network Manager may always choose wireless or wired as primary (based on other questions here) which seems to include interactions with the route's metric and dhcp, router advertisements (where NM followed a 1 metric), interface up/down and RedHat-specifics.

Assuming you're using it: DHCP may reset your route to the "better" one according to its route metric¹. Higher metric values are more "expensive" so the less costly metrics are preferred. In the Network Manager list here, it's noted that dhcp wireless metrics are penalized +100, and the NM developers say NM should just choose the right route, but also that metrics may be the best device method. I'm not sure this applies to your eth1, but you can control it:

Swapping the numbers (10 and 20) from this answer for your interfaces, I get:

$ sudo vi /etc/sysconfig/network-scripts/ifcfg-eth0
METRIC=20

$ sudo vi /etc/sysconfig/network-scripts/ifcfg-eth1
METRIC=10

That answer says this should prefer wireless and perform a graceful failover between interfaces.

If you want NM to select as the dev's may intend, GMambaG's interface keywords are also considered in the RedHat docs for static routes, which note unexpected problems using multiple gateways + GATEWAYDEV. To test that, start both interfaces and try to resume operations after removing eth1. If you have trouble (like intermittent timeouts), you should be able to recover by removing your changes and restarting networking:

$ sudo service network restart

If all else fails, router advertisements (ICMP types 9 and 10 I think) may be affecting your route tables; coordinate with the network admin before trying to block them.

Edit: Reduced this answer's text, placed OP's solution on top.

¹ If you're not using DHCP you'd have a manual gateway. This article suggests using NM_CONTROLLED="no" to prevent NM from automatically switching to that interface's gateway.