I'm British and I can't recall the term "government" being used in this way, even informally. I think most people, even if they're not interested in politics or our "unwritten constitution", would have some understanding that the courts, police, etc. are separate from the government that they elect. They may or may not have a word for what that thing is (the State) but nobody talks about people being "arrested by the government" or "convicted by the government".
I can imagine the people might think of the Civil Service as part of the government though.
In everyday speech, not really. We don't have a codified constitution so things are sometimes a little ambiguous. I guess people might use terms like "the state" or "the powers that be".
There is the legal concept of "The Crown" [1], which is kind of like the legal personality of the state. It sits at the root of a hierarchy, with the executive, legislature, courts, military etc. below it. The monarch, as head of state, exercises the powers of The Crown, although mostly in a ceremonial way on the advice of the government [2]. But nobody talks about The Crown in everyday speech.
[1] https://en.wikipedia.org/wiki/The_Crown
[2] https://en.wikipedia.org/wiki/Royal_prerogative_in_the_Unite...
In the US the Head of State figurehead role and the executive leadership are merged into the President of the United States, while a legislature remains separate. In the UK the Head of State figurehead role goes to a monarch (currently Queen Elizabeth II) but the executive leadership is granted to whichever group controls the legislature as Prime Minister.