Hello,
  I’m an American currently living in Spain with my boyfriend. He has a student visa, but I have been unable to find work (I teach English). My tourist visa is almost up. Is it possible for me to go to the UK for a visa run? If so, how long would I need to stay? 3 months? 6? Thank you for your help!

Hmm, not sure if the UK works for renewing, but in theory yes. And if you’ve been there three months then you need to be out of the Schengen countries for three months then.