Should I send a follow up email? I had a PhD interview nearly 3 weeks ago and it went well. I spoke to the supervisor after the interview a few days after and he said I was the candidate they want but that was 2 weeks ago. My question is, should I send a follow up email politely asking what the status of my application is. If it helps, I'm in the UK
I would be fine to ask. Thank them for their interest. Ask about the current status. Ask whether there is anything they still need from you. 

Three weeks is about right. If there is a problem then it is time for you to know.