Yeah, try changing the power supply.
Remember that this if a problem has occured, it might start up better and boot the computer up after being recently turned off since the capacitors still store power, and so it only needs a little kick.
Its the only thing that seems could cause the problem after you saying this:
"However, after having switched it off, if I switch the computer back immediately afterwards or within several minutes of doing so, it boots up through successfully and works perfectly"
I honestly do think it is that, so i'd try changing it, and if it works, then great