For your specific example above, is it necessary to pull the UK visitor from AM? You could utilize geo targeting within Test and Target segment creator to capture UK visitors. (if available for client) This way you could possibly create this segment without any integration.
You need to setup the integration of Audience Manager and Target. There are 2 ways to do this:
The new Share Profiles and Audiences core service in the Marketing Cloud.
The legacy AAM to TNT integration.
Once they've got the integration in place they can create a saved segment in Classic UI that include the AAM segment AND the mbox parameter.