The paragraph quoted is the condition for lounge access. As fast as the lounge for access by COS, it is governed by:
"Customers flying First or Business Class have access to the equivalent class of lounge regardless of their frequent flyer status. (First Class passengers may be accompanied by one guest travelling on a flight operated and marketed by a oneworld airline).
..."

There have been many discussions on whether a passenger can be admitted to any OW lounge of his/her choice by COS, but different people interpret the rule differently. BA is very strict, but I don't know about JL. She can try to see if she can get in. It won't hurt asking.