us citizens – What passport should I show my son who travels to the United Kingdom? American or British?

I am American, my husband is Brotish and my 3-month-old son has British and American passports. I will travel with him to visit my husband in the United Kingdom for 4 months without Visa, and then I will return to the United States to apply for the appropriate settlement Visa.

In the immigration of the United Kingdom I must show to my children the British or American passport? I'm worried that the immigration officer looks suspicious if I show a US passport for me and a British one for my son. Is there any law that says you must use a British passport while in the United Kingdom as in the United States? Does arriving with a US passport affect your ability to use the NHS?