If President Barack Obama was born in Hawaii, then could he be British?

His father was under British consular jurisdiction.

The requirement that for consular registration within one year of birth applied only if the father was a Citizen of the UK and Colonies (CUKC) by descent only and the child was also to become a CUKC by descent rather than birth in the UK or colonies. As Barack Obama Sr, was born in the colony of Kenya, he was a CUKC by birth rather than descent. Thus, Barack Obama II was capable of becoming a CUKC by descent without the need for consular registration.

British Nationality Act, 1948
(11 & 12 Geo. 6.) CHAPTER 56.

Part II
Citizenship of the United Kingdom and Colonies.
Citizenship by birth or descent.
5. Citizenship by descent.
“5.—(1) Subject to the provisions of this section, a person born after the commencement of this Act shall be a citizen of the United Kingdom and Colonies by descent if his father is a citizen of the United Kingdom and Colonies at the time of the birth: Provided that if the father of such a person is a citizen of the United Kingdom and Colonies by descent only, that person shall not be a citizen of the United Kingdom and Colonies by virtue of this section unless—
(a) that person is born or his father was born in a protectorate, protected state, mandated territory or trust territory or any place in a foreign country where by treaty, capitulation, grant, usage, sufferance, or other lawful means, His Majesty then has or had jurisdiction over British subjects; or”

Foreign consulates are authorized in the US under treaty, therefore “His Majesty then has or had jurisdiction over British subjects”.

By borderraven