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Generally, the person who files first claiming a child will get the exemption and the refund, unless the IRS has previously flagged the account. Once the second person files, the investigation can begin. If the first person did not qualify, the IRS will send a bill for repayment of all or part of the erronious refund.
Next year, when the person tries to file, the return will be flagged and blocked until that other taxpayer can prove she is eligible to claim the EIC.
You will get the refund only when the IRS determines you are entitled to it. I can't predict how long the IRS investigation will take.