Both parents should file amended returns listing the correct child. Depending on the ages of the children, there may be some change in tax.
For example, the child tax credit for children under 6 is $3,600 and drops to $3,000 for qualifying children ages 6 through 17.
Switching children will probably create a mess at IRS.
Both returns will not be processed together, so the first one processed with probably generate a notice saying the same child has been claimed on another return. It’s possible even if you explain the situation, the second return will generate a similar notice because it's in a different pipeline.
Consider going to a local IRS office together with both amended returns. They may be able to coordinate them.
How do I amend my 2021 return?
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