This is one of the topics that I studied in my undergrad for political science. You aren't going to find many expert historians on both the American and British political systems and even fewer willing to make generalizations. That's completely reasonable of course. There's a lot of nuance and complexity in these systems that evolved both independently and in dialog with each other. Historians, and I did study that formally in school though not enough for a full minor, are careful to emphasize particularity and contingency. That means that history isn't physics. There aren't any "fundamental forces" that can be called upon to explain everything that happens. There are both systemic pressures and individual choice involved. At its base, historians would say that the systems are different because of the specific processes and people that shaped them. You could write multiple books detailing all of that.

That said, there are some avenues for investigation. In the past, America had periods of being a multi-party state. Look at the 34th Congress' House for example. It consisted of 83 Democrats, 54 Whigs, 51 Know Nothings, 22 Anti-Nebraska movement members, 13 Republicans, 9 Indiana's People party members, 1 Free-Soiler, and 1 Independent. This was a time of great transition in the country as political coalitions collapsed and reformed around new issues. At the same time in Great Britain, there were only 2 parties; the Conservatives and the Whigs. So one can't claim that eternal and fixed conditions forced the two countries into the positions that they are in today.

One thing that I focused on in my research days was how the American political system evolved in the face of the rise of the power of the presidency. In the past, Congress was seen as the superior branch of government. Presidents were frequently called to task by Senators and House leaders. This began to change around the turn of the century. The presidency, an all or nothing political position, become the most important post to win for each political party. Minor parties that were still influential at the time, like the Progressive Party, had no chance taking that office. Having seats in the House or Senate was no longer enough in the eyes of most voters.

At the same time there was a major election in 1912 that featured three main candidates. The Republican party split in two between supporters of former president Teddy Roosevelt and current president Howard Taft. They managed to demonstrate one of the main weaknesses of third parties in winner take all systems, their opposing campaigns split their majority and gave the election over to their mutual opponent, the Democrat's Woodrow Wilson. This stark lesson soured Americans in a big way on third parties in general. In the years afterward, the spoiler effect would be brought up again and again to discourage voters from supporting the weak alternative parties that continue to exist in the United States.

In UK, Parliament only increased in strength over this period. The monarch became weaker and more deferential to MPs and especially the Prime Minister over time. In 1936, this would culminate in the government forcing Edward VIII to abdicate in favor of his brother given his impropriety with a twice divorced American and sympathies to continental authoritarians.

But what about British political parties? At the end of the 19th century the Liberal party, an alliance of Whigs and pro-capitalist Conservatives, and the Conservatives remained the only important parties in Parliament from what is today Great Britain. Obviously, there were the Irishmen pushing for eventually Irish independence but they don't really have an analog in the modern parliament. What we see instead in the next few elections is a rise of a more left-wing party in the Labour party.

The Labour party, unlike the various socialist and communist parties in the US, managed to build momentum. They didn't have to directly compete with the Liberals and Conservatives for a presidential office and could instead focus on the urban districts in which they had the most power. Rather than representing the interests of aristocracy like the Conservatives, or the bourgeoise like the Liberals, Labor took up the cause of the common worker. In a Britain with an expanded franchise and a thoroughly industrialized economy, that would prove to be a big winner. In the 1920s, they overtook the Liberals (who were actually in a period of infighting). In 1929 Labor won a plurality of seats, but no party had a majority and they couldn't agree on a coalition government with any other party. The Liberals would end up teaming up with the Conservatives to form a national government in 1931, and in many ways this broke the party. The Conservatives evolved in a more bourgeoise friendly direction and Labour welcomed many disaffected social liberals. The Liberal party clung to influence in fewer than a dozen local constituencies.

But in a parliamentary system, that can be enough to demonstrate relevance to voters. Liberals now had no hope in electing the next PM, but that didn't mean they had no influence. In several instances since that time, they have operated as king makers by swinging their vote to or from a government of a different party.

The rise of the Scottish National Party is an entirely different issue, and one with which I am less familiar. To me, it actually demonstrates the power of strongly sectional parties in a FPTP system. They don't have much national influence, but they only need to win local elections to have national importance. Much like Bloc Quebecois in Canada. This is simply not how US politics functions. There is no strong Texas National Party or the like.