Upvote:-2
With a British passport, you can go the the US visa-free. According to the US embassy:
Currently, many British citizens traveling on a valid, individual machine readable passport, with a return or onward ticket, and who are staying for less than 90 days, qualify for the Visa Waiver Program and can travel visa-free with only an Electronic System for Travel Authorization (ESTA).
During your 90 day stay in the US, you can collect your US passport. But if you do this, you'll need to exit the US with your British passport within 90 days, and come back into the US with your US passport when it's available.
Upvote:2
Under US law, you are required to enter and leave the US bearing a US passport, but there are no consequences for violating this rule per se.
Nothing will stop you from leaving the US in any case, since the US doesn't have any exit checks. Coming back to the US is the real question.
The proper way to come back to the US is with a US passport. You haven't said in your question why you can't get a US passport. If it's because you don't have enough time to get one before you travel, there are various solutions:
As a US citizen, you cannot be denied entry to the US if you can satisfactorily prove your US citizenship. So if you manage to make it to a US border with satisfactory evidence of US citizenship (e.g. expired US passport, US birth certificate with photo ID, Certificate of Naturalization, etc.), they will eventually let you in. The problem is how to get to the US.
Airlines will generally not let you board without documents that let you travel to the US. Without a US passport, what documents will you have to do that? You can't get a US visa (you will be denied). I can think of several possibilities: