There is not an appeal process for custom URLs.
Custom URLs are not really "custom." They are simply "pre-approved" URLs. Ideally, Google should have made it clear that your options are for what they offer ... not what you would like to request.
Those of you who had an option to request something different, are in the minority. That option was removed shortly after the Custom URL program was implemented and there is not an appeal process after Google responds to the request you submitted. (You can't use the Google+ Profile names appeal for Custom URLs.)
If you don't like the way Google has implemented the Custom URL program, please use the "Send Feedback" tool to let them know how you would have preferred things to work.
Do not accept any URL that you don't really want, as there is currently no way to change it (or remove it) if you change your mind later. Just periodically check again and see if there are new offers.
If you don't see a red banner at the top of the page, you can try going through Edit in the Links section of your About page to see what offers are available to you.