CPU is the heart of the computer and operating system is the main component required to do all our task in it. In times the system becomes older or due to other issues its losses its capability to do its task in a manner as it already does. Once your system is not performing well or it is dead you plan to switch your data from that system to another in such situation, yes you can transfer the data to another system once if you place your older hard disk as secondary.
You Can ask what will happen to the operating system in that older hard disk, will it work in the new processor or you wanna install a fresh copy of operating system ? Answer for this question is No you cant switch your operating system to the new processor unless it is a retail version of operating system.
Once you install a fresh copy of operating system it will create a boot sector associated with the processor and its hardware. If you change the hard disk to new system with different hardware and processor then the operating system boot sector for this processor will vary and it wont compatible with the older one. You can use your older hard disk as secondary hard disk with the newer one if it available. In other case your present hard disk is the only one avaliable then you want to install the fresh copy all your data on the c drive will be lost.
If your version of windows is retail then you will not face any problem with the operating system this is because OEM version of the operating system will work only in one system. But the retail version of the operating system will work with any number of system but at anytime only with a system. Hope this information might be useful for you, Happy Computing !