Yes, you should be able to start contributing to your Roth IRA as long as it is within the limits. Please see Amount of Roth IRA Contributions You Can Make for 2020 for details. The conversions from your 401k to Roth IRA are not considered contributions.
You will get a 1099-R for the conversion you conducted at the beginning of 2020 and it will be included on your 2020 tax return.
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