To elaborate on Patricia Shanahan's comment:
Nationality rules are complicated enough that it is not reasonable to expect officials of one country to reliably determine whether someone is a citizen of another country. The passport system allows each country to give its citizens a document that is sufficiently standardized that it can be read and relied on by officials of other countries. If you want US officials to treat you as a British Citizen, show them a British passport.
In fact, US law is explicit on this question. From 8 USC 1187:
... an alien who meets the following requirements:
(2) National of program country
The alien is a national of, and presents a passport issued by, a country which ...
Therefore, presenting a Polish passport will not qualify you for the visa waiver program, regardless of your other citizenship, and regardless of other evidence you may have of that citizenship.
As Peter Green notes in his answer, your can either get a British passport and use that for the VWP or get a US visa with your Polish passport.
As it's only two weeks before your trip, I would recommend making an urgent application for a British passport if you are in the UK. The cost (GBP 111 to 137, about USD 150 to 180) is similar to the cost of a US visa (USD 160, about GBP 120), but success is far more probable, and the usefulness of the British passport is probably far greater than that of the US visa.
You can look up US visa processing times at https://travel.state.gov/content/visas/en/general/wait-times.html/. The current times in London are 7 days for an appointment and 5 days for processing. Belfast is somewhat better, at 3 days and 2 days, respectively. The consulates in Edinburgh and Cardiff are not listed, so I suppose they do not process visa applications.
If you are not in the UK and cannot travel to the UK, you cannot make an urgent passport appointment; instead, you can apply for an emergency travel document. To enter the US with one of these, however, you need a visa, so there's no reason for you to pursue that.