Current Tado hardware allows control when internet goes down.
The Bridge (a small white box with 3 LEDs) has no connections other than power in and Ethernet socket. It connects wirelessly to the Thermostat to allow data interchange between the Thermostat and their Internet Servers.
The Thermostat can be hard wired to the Boiler as any normal stat but still need it's battery. To use the Thermostat wirelessly it has to wirelessly connect to the Extension Kit (a larger white box). The Extension Kit is hard wired to controls to switch both heating and hot water. It has to also connect wirelessly to the Bridge to allow Servers to control it.
I found the online installation instructions a dream to use if you tell it what your previous setup was. Extension kit just swaps onto a standard controller backplate.
Once installed Tado works really well, particularly the location detection using mobile phones to turn system off when we're out. I can also use my phone or Alexa to control it remotely.
Tado is a very German company, owned largely by Venture Capitalists.
The Trustpilot thread referenced to above is factually wrong. Tado v2 devices purchased outright are still fully supported by current and future Apps without subscription.
I have no connections to Tado - bought mine cheap on a well known auction site.