Why do British cars of the '50s look so different from American cars of that era?

Because the UK had just suffered the cataclysm of WWII and they were focused more on rebuilding their bombed and shattered country than on adding fins to cars.

IMPORTED CARR

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AMERICAN GAS!

They went to war in 1939, not 1941, The UK paused making cars in 1939, and the US in 1942. There was a lot of advances in car styling in the US between 1939 and 1942, so when both countries returned to civilian auto production in 1942, the US was ahead on the styling front.

Also, the US auto industry were firm adherents to planned obsolescence and annual body changes, to encourage people to keep buying new cars. It wasn’t quite like that in Europe at the time.

By the early to mid 50s, British cars started to look more contemporary, adopting US and continental Europe styling cues.

Obviously, because cool and pure edge didn't reach London until the 60s.