IRS will use the information on your 2020 tax return to calculate the amount of Advanced Child Tax Credit. Since the child's mother claimed the child on her 2020 tax return she will receive the advanced Child Tax Credit payments. This is just an advanced payment of the credit that would be given on the 2021 tax return.
If the mother doesn't qualify to claim the child on her 2021 tax return, she will have to pay back to IRS all or any amounts she received as an Advanced Child Tax Credit payment.
You will need to determine who can claim the child on the 2021 tax return to see who is eligible to get the Child Tax Credit. Here is a link to an article that lists the requirements to claim a dependent.
https://turbotax.intuit.com/tax-tips/family/rules-for-claiming-a-dependent-on-your-tax-return/L8LODb...
Whoever meets the requirements to claim the child, is the one who will receive the Child Tax Credit if all requirements are met (see the next link).
Here is a link to the qualifications for the Child Tax Credit.
https://turbotax.intuit.com/tax-tips/family/7-requirements-for-the-child-tax-credit/L3wpfbpwQ
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