Dual Citizen US/British with only British passport [duplicate]

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:

• Apply for a US passport in the US in person as a Passport Agency, which can issue it within days.
• Apply for a US passport in the US and have someone mail it to you when it's received if you're abroad at that time.
• Apply for a US passport at a US consulate while you're abroad.

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:

• British passport holders generally qualify to use the Visa Waiver Program. VWP requires getting an ESTA beforehand (airlines will check that you have gotten it). You are not supposed to be able to get an ESTA if you are a US citizen, but maybe it will work since it's an automated system (who knows?), if it is not very obvious that you are a US citizen (e.g. if you were not born in the US). If you can get ESTA, you can take a flight to the US, and then present yourself as a US citizen to US immigration.
• There are US preclearance facilities in various parts of the world, including Dublin and Shannon airports in Ireland, where you go through US immigration before boarding the flight. Here, you don't have the problem of the airline refusing to let you board. I am not sure whether the rule about CBP not being able to deny citizens entry also applies to preclearance facilities, but if it does, then you should be able to present evidence of US citizenship, and then go through and fly to the US.
• You can first visit Canada or Mexico on your British passport, and cross into the US on a land border. Here, there's also no issue of a carrier refusing to let you board. You would again present yourself as a US citizen to US immigration.

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.