How to recover my iCloud account without email access?

I just recently bought a new phone after my previous one went missing :( I know my iCloud username, but I unfortunately cannot remember the password, and the account isn’t connected to any of my other Apple devices. A verification code was sent to my iCloud, but I cannot login to it; I still have the password for my old phone. So, I'm able to pass that step and receive calls from apple but my dilemma lies here. It takes me to a page that states "We are evaluating your information to determine when you will be able to unlock your account. In a few hours, we’ll contact you at [my registered email] to let you know when your wait time will be complete. Apple Support can’t help you shorten the wait time. You can check the status of your request at any time at iforgot.apple.com." But I don’t have access to that email address, so how will they be able to contact me? I’m able to receive calls from Apple regarding my code—isn’t that enough proof of my identification? What should I do—call Apple Support?

[Re-Titled by Moderator]

Original Title: iCloud Account Recovery Issue