No, you do not need to calculate what you should have received for 2020 or file that. Yes, you will have to wait and file it all for 2021. The unemployment compensation taxation forgiveness for the first $10,200 does apply to 2021, as well.
The IRS requires income to be reported in the year the income was received, which is called the cash basis. The income counts when you actually get the money.
For what to expect for 2021 taxes, try using this calculator for a ballpark estimate: Tax Calculator and Refund Estimator 2020-2021 | TurboTax® Official. This could help you make sure you have enough taxes taken out along the way for the rest of 2021.
