There was no possibility of a peace with Nazi Germany that would have been anything other than a short armistice. Had Britain chosen to seek peace in 1940 following the Fall of France, then it is likely something resembling the Peace of Amiens between Britain and France in 1802-1803, and almost certainly uses the German invasion of the USSR in 1941 as an excuse to resume hostilities. Nazism had proven itself to be impossible to deal with: Hitler had broken treaty after treaty and had followed every act of appeasement by demanding more. By 1940 Hitler's Germany was no longer a power that could be trusted to keep compacts. Furthermore, and contrary to popular belief, outside of a handful of vocal exceptions, the British elite was at best ambivalent towards Nazism, while the British people were on the whole disgusted by it, especially after such incidents as the annexation of Czechoslovakia in 1939 in violation of the Munich Agreement, and the Kristallnacht pogrom against German Jews the previous year: the British Foreign Office, then still attempting to appease Hitler, openly admitted that Nazi Germany's violent antisemitism was a major stumbling block in finding an accommodation with Germany that would have been accepted by the British people.

Additionally, Hitler's vision of a Europe under the domination of a single continental hegemon, quite apart from the evils of Nazism, was antithetical to British foreign policy for over two hundred years. It's why Britain fought the Kingdom of France for a century, backed coalition after coalition against Napoleon for twenty years, then allied with France against Imperial and Nazi Germany when it replaced France as the most likely threat to the European balance of power. A single hegemon in control of all of Continental Europe's resources and the Channel Ports is an existential threat to Britain's security and trade. This is why it never accepted Wilhelm's or Hitler's offers of alliance, which were conducted in the assumption that Germany could have Europe while Britain could have the oceans.

As for Winston Churchill himself, his grasp of history meant that he understood early that attempts to appease Hitler were doomed, while he was especially revolted by Nazi antisemitism. No less a figure than Clement Attlee recalled that Churchill burst into tears when recounting to him the crimes the Nazis were committing against Jews as part of the state-mandated boycott of Jewish businesses after the Nazis came to power in 1933. The appeasers were discredited by the invasions of Czechoslovakia and Poland. The May 1940 War Cabinet Crisis was perhaps the closest Britain got to offering terms, when the Foreign Secretary Lord Halifax advocated for negotiating with Germany through Mussolini's Italy. However, Churchill's position as Prime Minister was secured when the Labour Party under Attlee made it clear that they, like him, believed that negotiations with the Nazis would be fruitless and that they were resolved to continue the war. The former Prime Minister Neville Chamberlain (and leader of the Conservative Party; Churchill wouldn't become leader until October 1940 when Chamberlain resigned the position, a month before his death from bowel cancer) sided with Churchill, effectively solidifying his support in Parliament. What little peace party there was in Britain was discredited after this incident and ensured that anyone who was likely to succeed Churchill should he, for whatever reason, had become incapacitated, would also support continuing the war.

Sources

Tim Bouverie, Appeasing Hitler: Churchill, Chamberlain, and the Road to War

Andrew Roberts, Churchill: Walking With Destiny

I think it helps to consider how discredited the appeasers / peace party became between September 1938 and 1939, and how war emerged as the only credible answer to Hitler’s actions, to partly answer this question from the British public / political side of things.

The appeasement position was fairly well respected in 1938, with most newspapers willing to give Chamberlain and the Munich Agreement the benefit of the doubt. By the end of 1938 this sentiment had turned, in large part driven by Kristallnacht (overwhelmingly due to moral outrage, but also an element of anti-semitism and fear of refugees). Opinion polling in late 1938 suggested that 40% were unhappy with Chamberlain’s performance and 86 per cent did not believe Hitler’s territorial ambitions were over. At this point, the public had turned in favour of an anti-appeasement coalition led by Anthony Eden (who resigned as foreign secretary in early 1938).

After the fall of Czechoslovakia, while there were a few pro-Hitler holdouts, British sentiment was firmly that Hitler could not be trusted and he was intent on taking over large swathes of Eastern Europe. The public still weren’t necessarily convinced that Britain should do anything about it. Instead, events were seen as a problem for “other countries” and there was view that Britain wouldn’t be bothered. In contrast to the German press (which framed Britain’s actions as encirclement of German interests), there was no consensus in the British press about what the government should do about the problem. Newspaper proprietors such as Max Aitken were anti-war, but more favourable to conscription as a clear signal of British resolve.

Promises to Poland were therefore fairly woolly and vague (e.g. “all support in [Britain’s] power”), and an agreement wasn’t signed until a week before the invasion of Poland. Instead, conscription was proposed in May 1939, and whispers of a guarantee for Poland began in the press and politicians. Initially, the Labour Party were against conscription, partly due to trade union concerns. They supported it with some assurances around job retention for men in training. Churchill and the anti-appeasers felt it didn’t go far enough. People were generally supportive of conscription (particularly working class people who saw it as an escape from their jobs), but there was no consensus for war at this point or strong guarantees to Poland.

The Molotov-Ribbentrop pact caught Britain by surprise, and by this point there was complete weariness among senior British officials. Alexander Cadogan diarised that evening that “we cannot go on living like this in Europe”. The government decided to tough things out, but Chamberlain’s stern message to Hitler was softened by pleas for a peaceful settlement. Newspapers and advertisements were equally incoherent - travel adverts exhorted people to “don’t mind Hitler! Book your holiday today!” as late as August 1939. Newspapers included curiosities such as “what to do with domestic animals in case of war?”.

By 31 August, with the first reports of war in Poland, fatalism had set in. Chamberlain gave a non committal speech in the House of Commons; Conservative Party anti-appeasers exhorted Arthur Greenwood (who was deputising for Labour leader Clement Attlee) to “speak for England” and press for war. Many normal people were disbelieving that there was war, largely because they hoped Hitler would stand down from Poland after Britain had shown its hand. Overall, though, people were accepting that war had come and was necessary. There was relatively little commotion when Chamberlain announced war on 3 September 1939, though some elites (particularly those with anti-Semitic views) retained their anti-war positions.

As noted here, the 1940 Cabinet crisis could be seen as a last gasp of some anti-war elites to retake control of foreign policy. But the reality was that the public was anti-appeasement by 1939 and war had emerged as the only credible policy alternative for the leading political parties. In that respect, the Norway Debate and Cabinet Crisis could be seen as the culmination of a long process of handing the baton from Chamberlain and the group who would soon become known as the “Guilty Men” to the anti-appeasement coalition led by Churchill.

Sources: “1939 - A People’s War”, Frederick Taylor; “Travellers in the Third Reich”, Julia Boyd