The local court can order anything they like however they cannot dictate what is allowed by the IRS especially when the it goes against the IRS rules. Sounds like this payment was considered part of the child support you are to pay and not deductible alimony(which is no longer allowed starting with 2019 tax year).
Let's start with the 1099 ... you could issue the nanny a 1099 since technically she is working for you as you are paying her fee. However if you are not the custodial parent according to the IRS rules (meaning the child does not live with you for more then 1/2 of the year) then even if you are allowed to claim the child because the custodial parent issues you an 8332 form you still cannot deduct the day care expense anywhere or take the child care credit.
