In AmE, we would use which, as the government or a department is a singular abstract noun. French does the same sort of thing, treating "la police" as a feminine singular, rather than a group of people in the plural.
In BrE, you will find it is quite common to use the collective singular as a notional subject or rather a metaphor for the plurality of individuals: "The team are happy to have beaten Celta Vigo." "The Home Office have made their decision."
So, it is perhaps only strictly correct to use "which" in such cases, but there will be people who prefer "who", especially if they are imagining the department's members as human decision-makers, etc., and are willing to bend the rules of grammar for this reason, as the Brits often do.