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This is most likely due to the account you are signing in with. Active Directory Users and Computers grants you access to Active Directory with the account used to open it.
1. Are you using the same account on both the new and old computers?
2. Are you running the ADUC as administrator(Right click run as administrator)?
3. Is ADUC running as a different user? As your domain admin perhaps?
I can't think of anything in group policy that would cause this.
Copy the shortcut and the ADUC from your old pc to your new PC.
If the account you are logging into your PC with doesn't have access to the AD accounts, then you may have been using another account to launch ADUC, maybe an admin account?
Right click on the OU and go to permissions and review what permissions the account that your signing in with has. I bet that maybe another account has the rights that you have been using on your old computer.
To do this, a domain admin would need to right click the OU that he wants to give you permissions to and select "Delegate Control". In there, he would give you the access needed by selecting your AD account, and then what control you need. This would resolve you issue with opening AD on your new computer and any computer going forward.