Two issues. First, the custodial parent, if maintaining a household for a qualifying child would be able to file as head of household every year even if not claiming the child as a dependent on the return.
Additionally, if there were any childcare expenses, the custodial parent can also claim the childcare credit every year even if not claiming the child as a dependent that year.
The only credit that moves from the custodial parent to the non-custodial parent in years they are able to claim them by court order is the Child Tax Credit. The child must be age 16 or under to qualify for this credit.
Second, is the stimulus payment issue. Whoever claimed the child on the 2019 tax return, assuming the child qualified for the Child Tax Credit (age 16 or under) would also have been eligible for the child stimulus payments ($500 and $600).
If the parent did not receive those payments they are missed at this point for that parent. There is no way for this parent to recover any child payments missed at this time.
The other parent claiming the child on the 2020 tax return, assuming the child qualifies for the Child Tax Credit, would also be able to claim the child Recovery Rebate Credit on the tax return ($1100) even if the other parent received payments based on the 2019 return.
Here is a link with more information on the Recovery Rebate Credit.
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