The taps of my shower are constantly hot

There are some shower mixers out there that thermosiphon even when the out let valve/ tap is closed. This could be caused by faulty check valves or design of the shower mixer. If your able to trace the cold pipe and heat is travelling along its length that would confirm it. If it is this that the remedy would be to change the check valves or fit a new shower.

From the sounds of it the thermosyphon valve is broken and not allowing cold through to mix with the hot so you can get your required temperature. If hot is travelling up the cold inlet pipe this will be the cause. To put this right I personally would recommend you have a new shower installed as depending on type and make of shower actually a repair can work out sometimes more expensive. A new Bar shower for example are around £80 plus VAT where as depending again on shower make and model after running round ordering parts the replacement can be cheaper.

The taps remain hot due to a flow of hot water going through them. This is almost certainly due to the shower valve being faulty, or plumbed in wrong, If the taps have always got hot since install then the later may be the case. Get a plumber to look and check