You should pay tax in the USA only on both pensions once you move there (but see below).
The state pension is nearly always taxable in the country of residence of the recipient as it is paid gross and it would be impractical for the government to chase pensioners around the world for tax. The state pension is covered by Article 17, paragraph 3 of the UK/USA double tax treaty here.
As far as your NHS pension is concerned, as you will read here, this is not treated as a government service pension (government service pensions are taxable only in the UK) if it is paid by CAPITA or The Paymaster General's Office. If it is paid by CAPITA or the PGO, it will be taxable only in the USA. If it is paid by the local health authority it will be a government service pension and will be taxable only in the UK (but see below). Occupational pensions and government service pensions are covered by Article 17, paragraph 1 and Article 19, paragraph 2 (a) of the UK/US double tax treaty.
There is a caveat on government service pensions. If your NHS pension is treated as a government service pension but you are a US national and you are resident there, it will be taxable only in the USA. See Article 19, paragraph 19 2 (b) of the UK/US double tax treaty.
Before you leave the UK, you should complete and submit a P85 to the tax office. This will alert HMRC to issue a NT (No Tax) code to your NHS pension payer, if appropriate.
I hope this helps but let me know if you have any further questions.