Not to mention that you have to actually get into the UK in the first place.
If you're moving to live and work in the UK, but you don't have a UK passport, then you will need to either prove that you are a UK citizen some other way, or you will need to have a work visa.
Or if you enter as a visitor (a visitor visa is 6 months, not 90 days), you will have to prove that you are not coming to live or work in the UK, that you have a return ticket, enough money to support yourself for the trip and you have a job, home and a life to return to in the US.
Chances are if you come to the UK with no UK passport, no other evidence of your claim to UK citizenship, and no ties to show you will return to the US, you will be refused entry and put on a plane right back to the US.
You might as well just apply for the UK passport now, in the US, and move to the UK as a UK citizen with no issues at all.