I was born in England to American parents.I have dual citizenship.I have lived in the US for 47 years.I now wish to move back to England. My child passports have been lost. What do I need to do?
You will need some documentary proof of your British nationality. According to the UK's citizenship checker, you can prove that you are British by showing that you were born in the UK before 1983. (See https://www.gov.uk/check-british-citizen/y/before-1-january-1983/yes.) Therefore, as implied by user102008, a birth certificate will do. Most likely, though, you'll want to get a passport.
If your birth certificate is also lost, you can find out how to get a new copy at https://www.gov.uk/order-copy-birth-death-marriage-certificate.
To apply for a UK passport while you are in the US, visit https://www.gov.uk/overseas-passports. There, you can choose "replacing a lost or stolen passport"; you will also find links to the application guidance and to a list of supporting documents you will need to submit.