Speaking from personal experience (LSE MSc Government) I don't think work experience is that important. I had summer jobs in government departments during my undergrad, but nothing exceptionally relevant - mostly just administrative. I don't know for sure, but I don't think that experience had anything to do with my admission because friends of mine also got in with absolutely no relevant work experience (retail, summer camps, etc.). So basically I wouldn't worry if you have no experience, but I'm sure if you have something really relevant it can't hurt.