There were a lot of British politicians who thought Britain should remain neutral. Even after the German invasion of Belgium many were reluctant. However, Sir Edward Grey, the Foreign Secretary, made a brilliant speech in the House of Commons, emphasizing Britain's effective if previously undefined obligation to France and the danger to Britain from Germany's invasion of a neutral country of strategic importance to Britain.
Wiki sez "many Liberal MPs, including at least a third of the Cabinet, and the Liberal press apart from the Westminster Gazette, wanted Britain to stay out".
The Cabinet voted to issue an ultimatum, which Germany rejected; the government announced a state of war. Two ministers (John Burns and Viscount Morley) voted against the ultimatum and then resigned.
There were also some further-left politicians who opposed war on general principles - for instance, future PM Ramsay Macdonald, who became a conscientious objector. Also, some of the more extreme Irish nationalists and republicans looked on Britain as the enemy, and Germany as "the enemy of my enemy", therefore, an ally.
The US position was somewhat different. The vast majority of Americans wanted to stay out of it. Wilson's re-election slogan in 1916 as "He kept us out of war." The vote to declare war in 1917 was very contentious, with about 1/3 of Congress voting "No". Opponents included both left-wing pacifists and conservative isolationists.
In 1914-1916, there was substantial sympathy for the Central Powers, especially among people of rececnt German and Austrian descent (including immigrants). There was also hostility to the Allies among Irish-Americans, who hated Britain. and among Jews who condemned Russian anti-semitism. I cannot name any political figures who explicitly supported the CPs in that period, but I'm sure there were a few.