No, they are American citizens by virtue of law and statutes. Did you have to fill out paperwork for them?
I know I did for my son born in Germany. He was not automatically a citizen, even though born of two American citizens at a Military Base. I had to have forms filed with the State Department.
German law is different from British law. The Germans have always gone for being of German blood rather then being merely born in Germany.
Read the title of the FS-240 "Report of Birth Abroad of a US Citizen" with that form you either applied (like I did) for a US passport or added him to yours when you went back to the States. Look at the documents required to get the FS-240, these same document's could have been used to get his passport instead. He was automatically a US citizen.
"The child's parents may choose to apply for a U.S. passport for the child at the same time that they apply for a CRBA. Parents may also choose to apply only for a U.S. passport for the child. Like a CRBA, a full validity, unexpired U.S. passport is proof of U.S. citizenship."
https://travel.state.gov/content/passports/en/abroad/events-and-records/birth.html