As you are aware if you do not know your YouTube channel's Google account's password (they share the same password) then you need to reset it. And for that you do need to have access to the email service which provides you with the email address your YouTube's Google account is called.
If this is a non-GMail email then you will need to take your lack of knowledge of the password up with the third party site.
If it is a GMail you will need to use "can't access your account?" at www.google.com and use the recovery options which I trust you have put in place. It is customary for all Google accounts to have at least a recovery email, or secondary email address which can receive notifications. Secret answers to questions are another possibility, as is a cellphone number.
Once you have accessed the Google account you have access to its YouTube channel be default.