If it's rebooting during the OS load, put it into the bios setup from a cold boot and then go to the hardware monitor screens. These should give you an indication of non-loaded CPU and MB running temps. And running voltages.
An overheating CPU will cause reboots. As will an overstressed PS or flaky RAM. Sometimes poor cooling management will push a system over the edge. Check your mainboard temps. If they're too high, then you have general cooling problems to deal with. An upgraded CPU or GPU can overload some OEM boxes not designed for the heat load. These same boxes also tend to have PSUs with insufficient rail headroom.
Any CPU work needs to be followed by a good cleaning/re-greasing of the cooler/chip interface, but you know that. Sometimes, with a CPU transplant, it is also a good idea to clear the cmos and then reset the configuration in the bios. Of course, you want to make sure that you have the latest bios to support any CPU upgrade . . . not that a 1 ghz Athlon is all that cutting edge, but I don't know your mainboard.
And lastly, driver issues can also cause reboots, and you seem to be getting reboots at about that time in the OS load. Make sure that the OS is not set to reboot on errors. Sometimes, if you have a sudden driver conflict, the blue screen data can be very helpful. I would also try booting into safe mode to eliminate all driver conflicts. If the system runs without reboots in safe mode, it's almost certainly software related.