Present French passport ( issued in UK) not acceptable to USA border control.
How can it be brought up to date ?

The object of a passport is to prove who you are in a foreign country. Your passport obviously doesn't do that so you will have to take it to the french consulate, find out why, and if possible have it corrected.

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Having French passport does not guarantee that you can enter the US, immigration officers also look through your personal data and have the right to deny entry if they see you may seek migration. You need to have proof of financial support, employment in your home country, etc.

When your passport was issued in UK, but you don't have valid visa to stay long term in UK e.g.. You are not student in UK or not having work visa in UK etc. US immigration won't let you in...