Here' some checks you can try. First turn off the mains supply. Then you need to get the box off the top of the valve. The one in blueshields picture has a screw at one end holding the cover on, get this out first. Inside find two screws which hold the electrics part onto the valve body take them out and remove the head from the valve. If yours looks like
see if you can find the rout to getting it's head off.
Once the head is off, you will see a spindle, note its position, and then see that it rotates freely. The best thing to do then is get a new head and replace the old one. How are you on wiring? It's straightforward, but if you don't feel comfortable with it get someone else to do it.
These valve heads fail often, and they are a cheap engineering exercise, so if they fail they nearly always benefit from a head change, sometimes you can get away with a motor change.
Replacements are available online, or from the likes of plumbcentre etc etc.
To answer your questions, the CH and HW would work if the valve was stuck in the mid position. The valve is about £70, but you may be able to get ahead for under £50. It might work with out the handle, but I suspect something failed and the handle was broken trying to fix it.