How do I transfer contacts from one iCloud account to another account and download to my phone?

Do these steps if you want to download a copy of your contacts from iCloud.

1.  Sign in to iCloud.com with your login credentials.
2.  Select Contacts
3.  Click the gear icon, then choose Select All to mark all of your contacts for export.
4.  Select the gear icon again, and choose ExportvCard to copy your contacts to a .vcf file. (Your contacts will still be available in your iCloud account.)
5. Now that your contacts are copied to a vcf file, you can import them into another iOS or mac device.

See if this ASC post can help: https://discussions.apple.com/thread/5436276