Distributions from a health savings account (HSA) must be used to pay
qualified medical expenses
, or the disbursements are subject to income tax and may be subject to an additional 20% penalty.
A list of qualified medical expenses can be found in IRS publication 502 (available here: https://www.irs.gov/pub/irs-pdf/p502.pdf)
Some dependent care expenses may be deductible if your child is disabled. For more information on this, please see IRS publication 503 (available here: https://www.irs.gov/pub/irs-pdf/p503.pdf).
If you child is not considered disbled, then payments for his/her care (including a nanny) are not considered to be qualified medical expenses and cannot be paid tax- (or penalty-) free using your HSA.