Like most trades its going to be around the £200-300 mark for anyone thats good, experienced and hardworking... maybe more depending on their location or the size and complexity of jobs undertaken ie the ones that nobody else wants or can do.
Look at what number the cylinder stat clicks off at when the water is up to temperature, this should give you a good idea of whether the stat is faulty or not.
Are you or anyone else at the property complaining of the water being excessively hot, if no then unlikely to be a faulty cylinder stat. Is the domestic water tank constantly trying to refill when no demand?
If you set to gravity then your water will always come on with the heating. If you have two separate valves then your system is most likely fully pumped.