How long does it take a leech to attach?
Apparently, the leech won't wait for you to swallow it, but it'll attach in your aerodigestive tract as soon as it's in it. This case study of children who drank leech-infested water explains pretty well (and in a rather graphical way, in my opinion) what happened to them. I'll copy here just an excerpt (emphasis mine):


  A seven year old girl was brought from a remote village (Kamardin Karaiz) with four days history of cough with blood stained sputum after drinking water from a pond. She also developed pain in her throat and was unable to swallow except for liquids. [...]
  Her coagulation profile was with in normal limits however her hemoglobin level was 9 g/dl. [...]
  On direct laryngoscopy a leech was found attached to the right pyriform fossa with its one end in the glottis which was removed immediately. The patient made a smooth postoperative recovery. On recovery the patient had no dyspnea, stridor or dysphagia and was discharged the next day on oral antibiotics.


This article also reports a historical case of a leech attached into the throat of a Pashtun man in India. The leech was removed with no consequences.

In short, leech ingestion will probably result in having a leech sucking your blood from inside your throat. Although removing it will probably grant a full recovery, it doesn't seem a pleasant experience.