No, because 1) Turing wasn’t jailed and 2) The Government doesn’t put people in jail; that’s the courts.
1. The legislature makes laws
2. The courts make judgements based on those laws
3. The police officers drag people into jail
All 3 are part of the government.
“The justice system is one of the three branches of the state. The other two branches are the executive, or the government, and the legislature, which is the two Houses of Parliament”
- The investigators and prosecutors are dependent on the government.
- A judge sentences, but doesn't carry out the punishment.
But certainly the initiative to investigate and prosecute Turing did directly come from the government. It was after all the PM who apologized for it.