I don't know the exact airspace you were in or around, but assuming airspace isn't an issue, you didn't need a clearance to fly the approach. An ILS is nothing more than a long straight-in approach. If the approach controller was too busy to give you an approach clearance, I would have just switched over to Tower and told them I'm 5 miles out for landing.
As a VFR aircraft in class G (generally, the FAF is outside of Class D), no one cares what you do. The only thing you really need in that situation is the clearance to enter the Class D airspace. Executing the missed is probably the wrong thing to do since it would still cause you to fly into the Class D airspace.
Now, if this happened on an IFR flight plan, and you weren't given an approach clearance, you would either have to enter a hold at the FAF (if that was your clearance limit) or continue tracking the localizer at your last assigned altitude (if your last clearance was to intercept the localizer). I wouldn't even execute the missed since that will cause you to leave your assigned altitude. Simply follow the localizer and tell the controller they forgot to clear you when there is a break in communication.