I’ve never needed to force routing, because I don’t have your two-network situation.
I’m assuming that you do have a wlan connection, and that you know the IP address of the router for that connection.
This is what should work:
Hold the mouse over the NetworkManager icon in the tray. Click “Manage Connections”
Select (click) the wlan connection that you are using, and then click “Edit”. You may be prompted for the root password.
Next, click the IPv4 tab.
When I do that, there’s a button that reads “Basic Settings”. I can switch that button to show “Routes”.
After that switch, I see a screen listing the routes. The initial list is empty.
Click “Add”.
In the first column (labeled “Address” here), type in “0” (without the quotes).
The second column (“Netmask”) should default to “255.0.0.0”, which is correct.
The third column (“Gateway”) needs the IP address of your router.
For the fouth column (“Metric”), enter “1” (without the quotes).
Save the results, and test whether you can now get to the Internet.
Hmm. You might have to do the same for the wired network. In that case, you will want
First column: “192.168.10.0”
Second column: “0.0.0.0”
Third column: the IP address for the wired router
Fourth column: “0” (without the quotes).
But also check the box “ignore automatically obtained routes”.
The reason for doing this for the wired network, is so that it won’t try to route Internet through the wired connection. I’m not actually sure if that will be needed.
I might have those netmask wrong. My output from “netstat -rn” gives:
Destination Gateway Genmask Flags MSS Window irtt Iface
0.0.0.0 192.168.254.254 0.0.0.0 UG 0 0 0 enp4s0
127.0.0.0 0.0.0.0 255.0.0.0 U 0 0 0 lo
192.168.254.0 0.0.0.0 255.255.255.0 U 0 0 0 enp4s0
which seems to suggest that you need 0.0.0.0 for the netmask for wlan and 255.255.255.0 for the wired.