Shouldn't the cold be cold and the hot be hot to touch?
Yes. Some how hot water is flowing from the hot through the new panel to the cold pipe when the panel is not in use.
Is the hot and cold pressures equal at other points in the house?
If the hot water pressure is greater than the cold (shouldn't be) then hot water will pass from the hot incoming pipe to the cold, through the body of the old mixer.
This would especially be true when cold water is being used else where. Any cold water use at another faucet would cause a lowering of pressure on the cold side at the mixer valve.
What happens at the mixer valve if hot water is used somewhere else? Does the mixer turn cold?
What is the temperature of the cold water at other faucets? Is it normal of is it warm?
Edit:
Just reread your original post.
this is happening with all the hot water outlets in the house
Does that mean that you get warm water from the cold side at other faucets when first turned on?
Sounds like you have hot water from the hot side passing through the mixer valve to the cold side, going back to the cold water input to the water heater and being recirculated. In effect you have a recirculation line to the bath. Great for having instant hot water at the bath but not good for what you want. It also appears that when you use your new panel a mixture of hot and cold water is coming in the cold side and a mixture of hot and cold is coming in from the hot side.
The hot and cold water is getting mixed in the old mixer valve. Don't see how the panel could work unless some provision is made for keeping the hot and cold separate inside the old mixer valve.