Why would I have an exposed nerve after having a tooth crowned/capped?

No exposed nerve. Just hypersensitivity. Maybe a finish line on the filling or crown is open and causing micro leakage. Could also be your tooth was badly decayed. I mean drilling a tooth is surgery and it hurts. Nerve is injured whenever a tooth is drilled on. Just like when you have knee surgery. The knee is injured by the procedure until you undergo therapy and heal and recover You may have a nerve that needs to be root candled.Talk to your dentist

After 35 years of being a dentist I have found that if a tooth has sensitivity after a crown of deep filling, and after waiting six months, if there is still sensitivity, the sensitivity almost never goes away. Remember, pain is almost always a sign that something is wrong. In your case your crown was done over a year ago and you probably had an unhealthy nerve in the tooth to begin with and the crown procedure caused additional irritation and inflammation, and now the tooth's nerve has had more than it could cope with. I advise you see an endodontist (root canal specialist), who will test your tooth to determine the state of health the tooth nerve is in. If may be that you might need a root canal treatment to rid yourself of your sensitivity.