I wanted to post my experience for any students registered at a US university but living in the UK. I am a full-time US graduate student spending a year in the UK to do research (university/professorial ties also make me an unofficial exchange student of sorts).
When I visited the council tax office in person, I was told I would have to pay council tax because I wasn't a registered student at a UK university. However, I had my university graduate affairs office draft a letter on official stationary stating that I am registered as a full-time student - which is true - I'm registered for thesis credit. They scanned this letter (signed by graduate administrator) & emailed it to me. Despite the fact that I printed it on an awful printer (the printing truly looked suspect), the council accepted the letter & I am now exempt from council tax.
So, just a note: even if the council says you have to pay, get a letter from your US university & send it to the council. It could be helpful to do this before leaving the US so you have originals of the letter.
Given the economy, I'm so happy this worked out with little hassle! Council tax is expensive...