Actually, according to every nanny on this site, each family is a separate employer and must pay minimum wage. Therefore, there is no separate amount. You pay 1.5x on the part you pay ($12.50), unless she says no to the overtime at that rate, and then you can negotiate to pay more.
The FLSA, however, says that you pay 1.5x on the total amount ($37.50), because the law is that you are joint employers, and the rate you pay is the rate you pay as joint employers. You do not each have to pay minimum wage separately as long as the worker always receives at least the minimum wage for each hour worked, but you are each required to pay the full hourly wage for all hours worked, and overtime is 1.5x that rate. Here is that regulation:
https://www.dol.gov/whd/regs/compliance/whdfs35.htm
However, I did find a letter from the FLSA administration that talks about exactly your situation, and it says that you can negotiate a separate base amount for a different type of work and pay 1.5x on that amount (so, your $20/hr single child rate, or whatever you guys decide it is).
Here is what that letter says (
https://www.dol.gov/whd/opinion/FLSANA/2005/2005_02_14_1NA_FLSA_TwoJobs.htm):
However, if the employer meets certain conditions, Section 7(g)(2) permits an employer to pay an employee overtime compensation at one and one-half times a different hourly rate than the employee’s regular hourly rate. If an employer wishes to use the Section 7(g)(2) alternative compensation calculation, the employer must satisfy the following four requirements.
the employee must perform two or more kinds of work;
the employer must establish a bona fide different hourly rate for those different kinds of work;
the compensation must be paid pursuant to an agreement or understanding arrived at between the employer and the employee in advance of the performance of the work; and
the compensation must be computed at rates not less than one and one-half times such rates applicable to the same work when performed during non-overtime hours.