"Verification codes cannot be sent to this phone number at this time. Please try again later." error when sign in to my iCloud account

I bought a new iPhone, but before that, I erased all the data from my old iPhone without first transferring the data to the new one using iCloud. Now, when I try to sign in to iCloud, I can't receive the verification code that's required, and I get the message:

"Verification codes cannot be sent to this phone number at this time. Please try again later."

How can I fix this, and what needs to be done so I can sign in to iCloud? I have already activated Apple Recovery.

[Re-Titled by Moderator]