I have a power supply on another MACHINE that is faulty. Could I simply unplug the cables from the PSU
replace the old one with the NEW one and then plug the cables back into the new PSU?

Will I be ok with using the old cables from the old PSU with the new one? The cable management in the machine
was very hard to perfect and I would like to not have to go through replacing on the power cables and playing around
with them.I assume we're talking fully modular PSUS here?
If the PSU is the exact same MAKE and model, and you're sure the cables themselves are OK, then yes.
If they're different, it's unlikely, if the PSUs are the same OEM and series then it's sometimes possible though.  You'll need to provide more info to say for sure.

Please note: do NOT do this until/unless you're positive it will work.  Even though the cables may look the same, the pinouts may be different.  You don't want to give your motherboard 12V where it's expecting ground, for example - you'll kill something, possibly several somethings.