Those pictures look like the AT&T Uverse modem, but I could be wrong since I don't have that system. This product is a router that you connect to the cable/DSL modem (or other modem device). Don't your entertainment devices already connect to the existing network? Is there any requirement from UVerse to keep the existing network? Swapping out the router may not be a good idea until you check with Uverse to see if there is any requirement to use their setup.

If you have your heart set on this system, you can connect this product to your existing router using one of the existing ethernet ports (keeping everything else the same); if you do this, you would create a whole new network in your location using this Google Wifi product while still keeping the Uverse network. Both networks would still share the same available bandwidth.

With regards to the Nest and Wenmo ability to connect, I don't have experience with them, but this is simply a router setup creating a new network. Whatever issues you had or didn't have setting up the Nest and Wemo on your current network would be the same when attempting to setup on this Google network using this product. This would apply to any devices you attempt to add to your home or business network and isn't dependent upon the router system you have installed.