Entering the US with US citizenship pending

This would make me a dual national if my application is accepted and processed; however, the status of my application has not yet been confirmed, so I am still only a British citizen. I'm yet to know if I've even been accepted for citizenship.

Applying for a passport cannot "make you a dual national" -- rather, having a country's nationality is a prerequisite for applying for that country's passport. If you are applying for a U.S. passport, then that means (you believe) you are already a U.S. national. A passport is a proof of nationality, but not having a passport doesn't mean you are not that country's national. Whether you are a country's national is determined solely by that country's laws.

Children born abroad to U.S. citizens, where the U.S. citizen parent(s) satisfy the conditions for transmitting U.S. citizenship, are automatically and involuntarily U.S. citizens at birth. It is not "claimed". From what you are saying, you are and have always been a U.S. citizen since birth.

According to U.S. law, a U.S. citizen must enter and leave the U.S. with a valid U.S. passport. So since you are a U.S. citizen, technically speaking, it is incorrect for you to enter the U.S. without a U.S. passport; this is regardless of whether you have applied for a U.S. passport or not. However, if you hadn't applied for a U.S. passport, there is a good chance that the officers at the border won't figure out that you're a U.S. citizen, because you were born in foreign country and you have a passport from a visa waiver country (so you didn't have to apply for a visa and be scrutinized there).

I am not sure if the CBP officers' systems are connected to passport applications or not. But if they found out about that, then they will know that you already are (or claim to already be) a U.S. citizen, and there will be problems.

Update: According to 7 FAM 085:

b. Although a consular officer may not issue a visa to an individual who has been determined to be a U.S. citizen, if a nonimmigrant visa applicant has a possible claim to U.S. citizenship but is unable or unwilling to obtain documents to establish that status, as determined by the post’s citizenship and passport officer, the visa officer may presume that the applicant is an “alien” pursuing a nonimmigrant visa application. If the presumed alien is found eligible to receive the visa for which application was made, the visa may be issued prior to the final determination of citizenship status.

So it seems to be okay to use a nonimmigrant visa (like tourist visa) or VWP to enter the U.S. while your U.S. citizenship determination is pending.

I think I may know the answer to my question. I called the US citizen and immigration office today (London office) and spoke to a very helpful man. He explained that as long as my status of citizenship has not yet been confirmed, there will be no problem travelling on my UK passport, as I will not be regarded as a US citizen yet. He said the only problem would be, is if a few days before I fly my citizenship does get confirmed, then I would need a US passport to travel to the states.

I have heard that US immigration will turn you away as a visitor if they believe you want to live in the US some day, even if you are going through proper channels and still retain ties to your current home. Let me see if I can find a reference. This forum is for folks who are waiting for "fiancé visas" and find visiting a difficult thing. It says:

The answer is yes, visits are allowed. But whether you will actually be admitted lies entirely in the hands of the officer questioning you upon entry. And face it, hearing you've got a sweetheart you're intending to/have married waiting for you in the US is going to set off a "he/she is just going to stay here forever" bell to that officer. It's your job to convince that officer that you have a life back in your home country that you fully intend, and will, go back to.

I suspect the situation will be very similar for you. Have paperwork with you about the job you hold in the UK and the car you own there and whatever else might make it clear you will be going back after this visit. There are no guarantees, any and all visitors can be turned away by an office having a grumpy day.

You might also have the opposite problem: once you're a citizen, you're only allowed to enter using your US passport. But you might not have the passport yet. That one I would take up with whatever consulate or embassy is processing your application.