Power, have you got it? Kind of.

Forgot to add. the power pack diagnostic button under the plug in to the desktop shows green when I hold it in, and there's no beeps to suggest it's any of the components inside. 

Further update. The led code of my power button is orange, ut I can't find anything to tell me what that means? 

are you sure it is not Overheating? CPU Pump or GPU Fan failure can cause Issues like you have.

That was my initial thought when it first happened though given how it cuts out I'd have thought it was related to the power rather than anything else?

If I turn It off on the plug socket, leave it for awhile come back and turn it on it will work, it boots up but then the monitor doesn't detect input, and the computer turns it's self of, displayin the orange led then will try to boot it's self up a couple of times unsuccessfully, and will become non responsive to presing the power button on the front unless turned off and on at the plug with a break between,

but you've already tested the PSU and it was fine.

try Clear CMOS and Power Drain first. unplug all Cables, remove the Coin Cell Battery and then hold Power On Button for a Minute. see if it helps. and if not, then try to Boot with Bare Minimum. remove GPU, RAM and disconnect all Drives and see if you get a Beep Code without the Components installed.

Edit:

Edited my Post a Bit.