The UK's gov.uk is done by the Government Digital Service, which used to be part of the cabinet office. In my view it's one of the great successes of the Cameron ministry (without breaking the rules on politics, I take a dim view of his premiership in general).

They focused a lot on UX. For example, they avoid dropdowns - "the select component should only be used as a last resort in public-facing services because research shows that some users find selects very difficult to use." [Source](https://design-system.service.gov.uk/components/select/).

It's worth reading their design principles: https://www.gov.uk/guidance/government-design-principles