English ivy has been considered an invasive weed for some time. The issue is that it is so prolific in landscapes (and, unfortunately, now in the wild) that it would be all but impossible to eradicate. It is definitely not something that you want in your trees where it easily reverts to a stage of growth where it easily goes to seed because it can't be controlled.
If being grown and managed--at least in more urban settings--it generally isn't such a problem, so I would be careful in approaching your neighbor, especially if he is responsible in the way he grows and handles it. Perhaps you may be able to suggest some alternatives. A couple on the native side that may work well are sword ferns, kinnikinnick, and wild strawberries. Ornamentals include low growing types of juniper, star jasmine, or pachysandra, a plant that does well in shady areas.