"... it's actually for a writing project."
So, this is just for a story you'e writing?
If a teenager has come over from America, with one parent dead, and the other parent is a UK citizen and living here, but has not been involved in their life until now, and is not on their birth certificate, are their issues?
In order to attend a British state school the child must either have a right of abode (eg be a British citizen or have ILR, the British equivalent of a US Green Card) or have some other status like being a refugee.
From your OP I assume that they do not have a British passport.
First question, how did they get into the country? As a tourist?
Secondly, which country was the British parent born in? If not born in the UK then that could prevent the child becoming a British citizen. Which country was the child born in?
I assume the British parent is the father. If he is not on the birth certificate then he needs other evidence such as DNA tests or Court Orders that show paternity (British Nationality (Proof of Paternity) Regulations 2006).
You can then register the child as a British citizen by descent. This will involve providing evidence of the British parent's status and also, since the child is over the age of ten, there is also a "good character requirement" to meet ie they don't have any serious criminal convictions.
Once this has been done, you can then apply for a British passport for the child and then you're good to go.
Either that or come over as a tourist, and then just don't ever leave the country. Schools in the UK do not check the immigration status of pupils in their school, that's down to the parents being honest.