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I really like the work of the UK's GDS (government digital service)

Me too. Focusing on digital services was, and remains, one of the few things the UK government has got right in recent years, IMHO - and excelled at.

Is it anything to do with the government per se, or is it driven entirely by civil servants?

Francis Maude was the MP responsible for a lot of that. He was the only politician I ever heard talk about lean practices in software.

Yeah -- look into Francis Maude, he's broadly responsible for the current gov't focus on (good) IT practices.