I'm trying to download Diablo II from the Battle.net website, I know how to do this as I've done it numerous times before.

Once I have downloaded the program there is no option to 'install' Diablo II, there is only an option to play it loike it's already installed.

I've gone through my computer and cannot find my original Diablo II folder and it isn't visible in the Add/Remove Programs list.

However I do have private servers installed which are using the same files with a few changes, do I need to remove these in order to re-install Diablo II?

Remove everything, check your registry for remains using RegCleaner.

yep it's due to the private server of Diablo II, had the same problem long time ago.
Just delete everything related to Diablo II and reinstall it, after you can copy paste the private server again without problems.
GL