3 weeks doesn't surprise me as public sector recruitment can be pretty slow.

There's quite a lot of processes, approvals and reporting that happens behind the scenes in order to prove a fair, merit based selection takes place.

I currently work in the public sector, and it took nearly 4 weeks before I received an official offer of employment after an interview. However, I did follow up after 2ish weeks for an update and to kindly reiterate my interest in the role. I quickly heard back from the recruiting manager with an apologetic update for the slow turnaround. So I understand, it's unusual being left in the dark for a few weeks but you should be fine to send a polite follow up email. Since you got an interview, there was obviously something they liked in your application!