Not guaranteed to work but worth a try...
Download, install & run the free Belarc Advisor click here. The report it generates will give a list of all the installed software & their licence keys (lower down the page), including the McAfee one hopefully. Make a careful note of the McAfee licence key.
Download the same McAfee trial software (identical version too if possible) & install it on the new PC. Now try registering the licence key, you noted down earlier, with this new install. This option can usually be found under > Help > Register (or similar) of the McAfee software.
If it works, make sure to uninstall the McAfee software from the old PC, as you're probably not eligible to use it on two systems simultaneously.
If it doesn't work, then it's quite likely that the licence was only meant for the old system it was preinstalled on. Or, the versions don't match exactly.
Another thing to look into is the possibility of getting free security software for the new PC from your bank if you bank with them online. E.g. Barclays bank offers Kaspersky Internet Security (includes antivirus) for free to its online customers click here. G