You have tried the most likely fix, if you have a multimeter you could check the element resistances, I think there are three elements on these showers and the resistances should be between 12 and 40 ohms. (remember to isolate the mains power, etc)
Otherwise it sounds like one of the elements isn't getting power which could lead to a dodgy relay or triac. This can be checked by measuring the voltage across each element when the shower is on full as there will be zero volts across the element when it is turned off and 240 V when the element is turned on, obviously I don't know what your competency is in working with live electric so please get this done by a competant person if necessary. This will tell you which circuit is faulty and then you can replace the triac or pcb (relay) accordingly. From what I remember, the triacs are assembled to the top of the heater can on posts.
If it isn't this then it could be a bad wire connection somewhere or bad solder joint among other things, which take longer to find and sometimes don't warrant the time needed to find them. The multicore cable connections need to be checked. Another area is the flow valve, the motor or valve could be broken, can you see the flow being visibly controlled, if the flow is on full all the time it will be cold, especially at this time of year. Maybe by turning on the bath tap you are reducing water pressure to the shower and reducing flow resulting in hotter water.
I hope this helps, good luck.
Paul M, January 2009