Get that valve fixed, and the Tado would be perfect for you. As you only have a single valve in the CH flow, your system would appear to be configured for gravity mode. The wiring options for A two zone controller are a little different in gravity mode, and there are further differences between CH via pump, or CH via a valve. The valve has its own relay which closes when the valve opens, and this used to signal the boiler to fire for CH. In this instance, the live from the controller for HW ON goes straight to the boiler to fire it, the live for CH ON goes to the valve, which opens the flow and the closes the relay to fire the boiler. I'm curious as to why you boiler fires for CH if the valve is stuck. If stuck open, it would
keep firing the boiler continuously, but if stuck closed, then no CH.
It's possible that you Climote's two seperate zone live outputs have just been connected together to fire the boiler. In this case a live from the HW terminal is going to open the valve as well as a live from the CH terminal. This is why the valve has a relay, so that it can send a live to the boiler to fire it, but if the boiler call terminal is already live from another zone calling it (HW or CH2), this live will not feed back to the other zone and open it's valve. This logic is lost on some installers, who will happily join 2 or 3 zone ON terminals at the boiler, without realising this turns the system into a single zone, with any timer turning on all zones.
You need to check if that zone valve is opening, has it been mechanically locked open in service mode or is it stuck, and are there more than two wires connected to it, meaning it has an ON relay to fire the boiler. Get this sorted first, get two independent zones going, then you can consider better 2 zone smart stat systems.
I'll get it fixed when I get a chance, basically take off the motor part and check to see if the valve itself is able to operate/observe if the motor activates when activating the zone.