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 (this PDF, page 3):
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.