Without seeing the actual order, it is hard to tell how a Judge would react to the unilateral decision to care for your daughter yourself. I am assuming that you typically go to work at 2 p.m.? I agree with Jill that the Courts believe it is far better for the child to be cared for by a parent than any other third party. Mother is wrong for insisting on the use of a nanny when you are available. From a strategy standpoint, you should proposed a mutual modifiacation of the order to Mom and try to get her unreasonableness in writing. That way, if you end up in Court, you have the proof. In the mean time, I think you care for your child every moment you are available, rather than using a nanny or any other 3rd party.
In general terms, courts prefer that children be with a parent whenever feasible, beyond any other third party care provider, including other family members and/or step parents. This doctrine is known as the "parental preference." If you do not have this doctrine recognized in your court paperwork, or some other right of first refusal to provide care (meaning the child is offered to the other parent before any other care provider), then you should bring a motion to modify. It makes no sense for your child to go to a nanny when you are available.
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I agree with the prior two answers, but don't forget that you also have the ability to deviate from the order if the two of you can agree to the terms. If you and Mom can reach agreement, even if it's just through an email exchange, there's no reason to think anybody will complain. The Court certainly won't. In fact, if the Court sees the two of you working cooperatively together for the benefit of your child, you both get kudos for it.