IntheBorders has it correct, if you haven’t got good insulation then you will consume a lot of whichever fuel you are using.
The problem with heat pumps in poorly insulated houses is the amount / size of radiators that are required, and this is where the whole myth starts as I suspect not enough design is done or corners are cut as the install becomes expensive due to the work required to get the system to work on low flow temps.
if your load is 20kW that doesn’t change, what changes is the how to deliver that with low flow temps, without getting that bit right an ASHP will be expensive. If you get that right and understand how to use it then it will be similar in running costs to natural gas.