You report unemployment income in the year that you receive the payment. It doesn't matter when you applied for it. If you did not receive any payment in 2020, and you did not get a 1099-G for 2020, you do not report anything on your 2020 tax return. The payments that you receive in 2021 will be reported on your 2021 tax return that you file next year.
Note that some states don't mail out Form 1099-G. You have to access it on the state's unemployment web site. You might want to check on the web site to make sure that they did not issue a 1099-G to you for 2020.
