Having watched both the American version and the UK version, they're like two different programmes. The American one is funnier (edited and produced to be that way), it seems almost comedic in some ways and it's also a lot more dramatic with lots of dramatic music and thudding heartbeat like sounds. The narrator also drops in a lot of phrases like, "What happened next was going to shock everyone......" It seems to be more entertainment than documentary.
The UK version is a lot more low-key, though in recent series they seem to have been choosing some people for shock-entertainment value, which has made me start liking it less. I much preferred the first two seasons to the last two - it seemed more natural and genuine.
I think the shows also highlight the difference in the UK and US delivery norms. From the show it seems like most women in the US get epidurals, whereas it seems a lot less common in the UK, where active labour techniques (up and about, birthing pools, birthing balls etc.) seem more common. I also found it fascinating to see that in the US the obstetrician comes to deliver the baby and they always set up a table of implements. In the UK it's a midwive and all she does is put a sheet on the bed. I was also amazed by the luxury in the US version, the rooms are AMAZING!
It's really interesting to see both and both shows had me hooked, till I had my son! Now I prefer the US version, but only because it's so different to my own experience in a UK hospital, that I can't really compare them.
I really suggest watching both, it's good fun.
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