No you don't have to reinstall the Operating system on any server or operating system if you install a second or new processor. Here are the directions I have typed up in the Forum for a Multiprocessor install for win2k. There is also a Winnt4.0 Multiprocessor install in the FAQs. Please refer to it for further help.
Here is the W2k upgrade for a second processor:
. If you are in Win2000 and want to upgrade to another proc. do this:
To go to multiprocessor, the Processors have to be the same size/speed/stepping. Look on top of your processor, it has a whitish grey stamp on top which should have the speed/size of cache/stepping on processor for example: 80523PY350512PE SL2U3
The 350 after the PY is my speed the 512 is my cache and the SL2 is my stepping on the processor. Make sure you have the same speed/cache and try to have a stepping that is only one number off or the same anything else won't work.
Now for the Processor Upgrade this is how to approach it:
1.Upgrade the system BIOS Firmware
2.Go to F10 and go to Advanced Mode by pressing(Ctrl+A)
GO to Sys config>Advanced Mode> View and edit details>Advanced Features>APIC-Full Table> Save and Exit.
3. Down the server.
4. Add the second proc.
5. Get on to W2000 and then go to My comp>Manage>device Manager>Press+by my comp>ACPI>Driver>update driver>NEXT (Wizard)> Display a list>Pick ACPI Multiprocessor>Enter and youre done.
Good Luck.
I Hope this helps.