I thought I remembered Linda Colley writing about this in Britons (link: https://www.worldcat.org/title/britons-forging-the-nation-1707-1837/oclc/57382072), and she does in her chapter on Peripheries. My answer relies on the work she has done, as well as experience as an academic historian of eighteenth-century Britain (though not of politics!).

The War of Independence was controversial in Britain - and the debate in 1775 was more about whether Britain should declare war on it's own colony. Because of that, there is much better evidence of people opposing the war, rather than supporting the American colonists. I don't know if advocating for not going to war with the Americans counts as 'supporting' them as per your question?

We have quite a lot of evidence on just how divided Britain was on the issue of declaring war. Unusually, loyalists started to produce petitions with signatures to demonstrate support for the war. This started in Manchester, and similar petitions cropped up throughout the country after September of 1775. There were also petitions against the war which were most successful in places where lots of people had emigrated from to America: East Anglia, Scotland and Wales. In 1778, an anti-war petition from Norfolk got 5,400 signatures.

Into the war, being openly pro-American wasn't popular but existed. There are newspaper reports and affidavits that demonstrate pro-American radicals were beaten up in England. John Wilkes made speeches in Parliament in favor of the American Constitution which he called "The best constitution on Earth". Wilkes was not the only radical, but previously those of a republican bent could still play the patriot because they supported wars against Catholic powers such as France and Spain. Actively advocating for the American revolutionaries could mean driving oneself just too far into unpopularity to be an effective radical in Britain, and the position was seemingly uncommon.