new immersion heater has blown 15a cartridge fuse after about 3 weeks

Hi Stublack You've unfortunately got a short circuit on the circuit, cause the fuse to blow. call back the tradesman who fitted the heating element, it should be under warranty if it is the heating element which is at fault. If there is an isolating switch, you can determine yourself what is at fault, by turning off the isolate; which isolated the heating element off, if the fuse doesn't blow now then this is an indicator that your fixed wiring from the consumer unit to the immersion heaters isolator is good and the fault is from the Isolator to the Immersion heater. kind regards Kilowatt Electricals

I think it is most likely to be short circuit on the connection to the element. If the switch was not touched during the change of element this is even more likely. It could also be a faulty element but this is less likely in my view although possible.