Hi
Having worked with the Met Police I think I can give a little insight into the background checks.
Basically when you fill out the application form you have to put down the name and address of your parents and/or your parents significant others. The police check the information that you give them (and I suppose other onfo that they find) and that determines whether you are accepted. If it's not your cousin that had the record, it's most likely his mom or dad, or could even be someone else in the family. They always state that they cannot disclose the info to you so unless you press your family members, your cousin won't really stand a chance in finding out. Bit of a joke but that's the way it goes. Hope this helps.