It might be worth asking on r/economics, but since I studied some economic history, I'll try and give it a go. I did actually write my undergraduate history dissertation on the influence of Keynesianism on Gordon Brown's government, but that's after the 20th century.

So, first, what do we mean by Keynesianism? Broadly speaking (in a a basic sense, but this is a history sub not an economics one), it means using "pump-priming" to solve a recession. Recessions are caused by lack of demand = demand needs to be increased = the government should increase demand through public spending to give people jobs so they can spend money= the government should borrow money to do so.

Now, increasing demand leads to economic growth and reducing unemployment, as businesses sell more and employ people to do so. But it also leads to inflation, as they increase prices because they can.

Now, Keynesianism ended up being rather politicised. Increase spending on public services to give the unemployed jobs, and economists say this will pay for the borrowing you need? Labour thought this was brilliant. Give the underclasses make-work so as to do stuff that will cause the savings of the upper middle-class to reduce in value (ie inflation)? The Conservatives thought this was awful. Or rather, they came to believe it was awful.

Keynes' ideas were forged in the great recession, a primarily demand-based recession, and they worked rather well, though they did cause inflation. You can see the rise in inflation in the late 30s here: http://www.economicshelp.org/blog/2647/economics/history-of-inflation-in-uk/

To summarise: Keynesian policy raises growth and reduces unemployment while causing inflation. In a recession, that's worth it.

Now, after the end of the War, Keynesianism was still popular in the Labour party, and to a lesser extent the post-war Tories. So, government spending continued to be pumped into the economy, to create full unemployment... even when inflation was already high and there was no recession. This (among other factors) led to economic instability, and constant booms-and-busts. When the oil price rose in the 70s because of OPEC, that was a supply-side problem, and so Keynesianism was not a great way to deal with this. You can see how high inflation rose in the 70s from the chart.

This, among other reasons, led to some young Tories looking for other economists whose ideas were more palatable to them. Hayek and Friedman, primarily, who espoused supply-side reforms that, coincidentally, did not require the government to spend much money on the poor. Thus, Maggie and Co completely turned against Keynesianism.

But with the recession of the early 90s, some young Labourites, such as Gordon Brown, turned back to some Keynesian ideas, though perhaps not in such a deeper way.

However, saying all this, it's very important to note two things.

1. Keynesianism is one factor in these economic events, and not at all the only one. There was a lot of business with shocks, currency value, etc.
2. As I've tried to explain, for a lot of politicians support of, or condemnation of, Keynesianism, was far more about politics than economics. And they were often not actually very familiar with Keynes' work: I gave a very basic explanation of Keynesianism because that's about all that most politicians knew. This meant that their pursuit of "Keynesian economics" was often what they thought/wanted Keynesianism to be, and not what it really was. Similarly, the Conservatives dislike of Keynesianism didn't stop them from, occasionally, say, lowering taxes and not lowering spending, which also increases demand. So Keynesianism did have some influence on the British economy, but not as much as you might think.