UK citizen Entering the UK on a one way ticket and US passport

To expand upon my comment, your son is a US national who can enter the UK on his US passport under Appendix V of the rules, or in common parlance, a visitor (like a tourist).

Unlike the US, the UK allows a citizen to enter under whatever passport they have. British citizens use Paragraph 12 of the rules. Other people use different rules. Visitors use Appendix V.

So upon arrival, your son will have a landing interview and his US passport will be stamped with 6 months 'leave to enter' (as if he were a tourist). This sort of thing happens all the time and the Immigration Officer will know what to do. Bring his birth certificate accompanied by the parent(s) listed on the birth certificate.

And Oh! Visitors are normally asked to produce evidence of an onward journey, hence the comments about return tickets. It's certain that the IO will get worried about it. Your son will need to think up a really good excuse for not having something to show.

The Right of Abode avenue is a red herring at the moment. Ignore it. While it's true that your son might be able to acquire an ROA stamp in his US passport, it would be very strange for an American to do that. Most people just apply for a British passport. Also, you got dodgy information, a year? No way. Like I wrote in my comment, two weeks is a bit shy, more like a month for an in-country application for a first timer.

You described a two week visit, if you can stay longer, then apply for his passport. Fill out the form and take it to the Passport Office. First time applicants cannot use the express service BUT he can have his interview with the examiner and they will make copies of his US passport and return it to him in 30 minutes max. They MIGHT be able to mail it to him in the US when it's ready, I don't know if first timers can have a passport sent abroad, but it's worth asking.

Adding...

Calling the British Consulate in SF, or the British Consulate General in NYC, or the British Embassy in Washington is a waste of time.

Admission to the UK is only guaranteed if you are able to prove you have right of abode of the UK. In most other cases, the immigration officials at the port of entry have ultimate discretion and you have no right of appeal. Cases where you do have right of appeal include cases such as asylum seekers and those who have had British citizenship revoked, though obviously this doesn't apply in your son's case.

The reason why right of abode matters for admission in general is because it makes the question of "what if they don't leave when they say they will?" irrelevant—because they already have the legal status to reside in the UK, it doesn't matter if they do stay.

Therefore, admission will depend upon the discretion of the immigration officer, as your son will be unable to prove right of abode of the UK, as he does not have any of the required documents, per the above link. The fact he can claim to the immigration officer he is a British citizen is irrelevant. As such, all the normal rules about entering the United Kingdom as a visitor apply (ability to support themself, liability to overstay the allowed period, etc.).

Entering claiming to apply for a UK passport while staying for two weeks will be met with extreme scepticism: it takes six weeks to get your first UK adult passport, and he'll have to send the passport he's used used to come into the UK as part of the application. As such, how does he intend to get admitted to another country when he leaves the UK after a fortnight, while his US passport is held by HM Passport Office? This makes him sound like he's likely to stay longer than he claims, which is absolutely a bad idea if you want to get into a country!