Laptops are no different at all to desktops.
except, as you say, for example, the PSU which is not an easily swappable module, and that they run on an internal battery.
Or, to put it another way, they ARE different.
No different to a desktop with a 12V input supply on the motherboard (yes, those exist). The battery is relatively immaterial. If you cannot handle a laptop, you don't know enough about desktops, either.
There's no need to be rude, Monkeh. Laptops, as you know, are more complex to diagnose than desktops, primarily because so few of the parts are user-replaceable, plus they tend to be less generic / more proprietary, so people will not tend to have swappable alternative components readily to hand.
Back to the original problem, the OP said that she had bought a replacement motherboard that would include the CPU. Now you have her replacing the CPU and buying heatsink compound... why? If you believe that there is a fault on the replacement motherboard then she should return that for a refund / replacement.
Back to the point re the power supply - if the motherboard has already been replaced then the PSU is a reasonable next step for investigation.
It is built in on a laptop.JohnD, is the psu not attached to the motherboard? Is there any way of testing that?
No it's not. The PSU is the separate "brick" that feeds the laptop. PSUs can and do fail and it's a reasonable component to investigate.