US military children born in England

Hello. My question is:

If two American citizens were based in the UK for World War II or possibly some years after (not sure of exact dates yet) and the mother gives birth here, are there any situations where the child can apply for UK citizenship?

Thanks for your help.

If the children were born in the UK before 1983, they were born British citizens otherwise than by descent (Section 4 of the British Nationality Act 1948).

Their children would be British citizens by descent, wherever in the world they were born.

Thank you Simon for your reply. I wonder if...

"Provided that a person shall not be such a citizen by virtue of this section if at the time of his birth—

(a)his father possesses such immunity from suit and legal process as is accorded to an envoy of a foreign sovereign power accredited to His Majesty, and is not a citizen of the United Kingdom and Colonies"

...could apply to American military personnel stationed here. Hmmm. Interesting.

That language applies to accredited diplomats and not military service personnel. It's well known that the children of foreign diplomats who are born in UK, are not entitled to the provisions of BNA. However, I've seen references to Americans with British nationality born within the premises of RAF Lakenheath and Mildenhall.

That's great. Thanks for your help Sunburn.