Note: Present and former residents of the United States do NOT need to submit any U.S. police certificates.
https://travel.state.gov/content/travel/en/us-visas/immigrate/the-immigrant-visa-process/step-5-collect-financial-evidence-and-other-supporting-documents/step-7-collect-civil-documents.html
Yes, you must obtain one. You can find instructions under UK -> police, court records:
https://travel.state.gov/content/travel/en/us-visas/Visa-Reciprocity-and-Civil-Documents-by-Country/UnitedKingdom.html
Just to clarify, K1 applicants do not submit any documents to NVC. There is no interaction with NVC besides them holding your petition until the consulate requests a batch yours is in. You will bring the police certificate to the interview.
*** Topic moved from NVC forum to K1 Process & Procedures ***