Good evening Carolyn,
Initially it does sounds like the heat exchanger could be at fault, however this wouldnt give you bursts of hot \ cold water.
I am inclined to think this could be due to a build up of sludge in the system blocking the flow of water etc.
Run the boiler with central heating on and see if the radiators get hot, this goes a long way to prove its not the heat exchanger.
Sorry for the delay in responding.
The majority of cases you would expect to see a fault code if the heat exchanger wasnt working.
Reading the service manual here, on page it explains how to clean the HW exchanger, if this hasnt been done and no improvement noticed I cant see how they can say its the heat exchanger, I ma also more confident after sleeping on this that its not the heat exchanger but most likley the flow regulator.
There are two parts which are far more likely to be at fault (both listed on page )55:
60 DHW flow regulator 78138 61 and DHW flow sensor 78137
Though again if these failed it would generate a fault code, these most certainly would affect how much hot water was produced and regulate the temperature, these components can also be manually bypassed to prove if that is the fault, so I would be included to call the engineer back, get him to show you the fault on the heat exchanger which is causing 'erratic' behaviour, secondly ask him to bypass the sensor and see if the fault clears, then do the same with the regulator.
No problem Carolyn,
It appears the link didnt work, the service manual for your boiler can be found here.
Appreciate if you can rate the service received, and I will await to hear back from you.