Using money from an early withdrawal from a 401k to buy a house is not an exception to the early withdrawal penalty. There will be a 10% early withdrawal penalty as well as ordinary taxes if you are under the age of 59 1/2 if you took money out of a 401k to buy a house. The rules are different when you take an early withdrawal to buy a house from a traditional IRA.
However-----you say you took the money out in 2020 and now you are using the money to buy a house in 2021---maybe you had better explain more about what you have been doing with that money and why you withdrew it from the retirement account in 2020.
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The purchase of your first home has no bearing on the tax treatment of the 401(k) distribution that is reportable on your 2020 tax return. If the 401(k) distribution qualifies as a Coronavirus-Related Distribution, you can indicate to TurboTax that it is COVID-19-related so that TurboTax will include it on Form 8915-E to eliminate the early-distribution penalty and allow you to spread the income over 3 years.
The exception to the 10% penalty for early withdrawal does not require any reason other than that you suffered a COVID-related financial hardship. But that exception only applies to withdrawals made before December 31, 2020 and was not extended to 2021 by Congress.
401(k) funds are not eligible for the "first time homebuyer" exception to the 10% early withdrawal penalty, that only applies to IRAs. IRAs and 401(k)s are controlled by different laws and regulations even though they have some similarities.
The withdrawal is always subject to regular income tax. You will also pay the 10% early withdrawal penalty unless the withdrawal was made before December 31, 2020 and you suffered a COVID-related financial hardship.