I completed my undergraduate law degree in the UK and am thinking of pursuing an LLM in the US, focusing on human rights and international development. What are the best schools in the US for this?

Northwestern University

may I also suggest Notre Dame; Emory if you are more into Law and Religion (and related legal issues)

I'm from American University and our LLM in International Legal Studies has a stellar reputation in the field of HR law. You may want to take a look. They offer Spring admissions as well as Fall which is helpful for students just applying.