Which is the best non preservative tear drops brand? How much time will it take effect in the swollen cornea to make it smooth and normal?

Most are fine. The basic mix in most tear replacement drops is similar. There is enough difference that many will find one brand that they like the most. Without preservative, there is less risk of toxicity or allergy to the preservative for those who need drops chronically. If the cornea is swollen, that is more of a problem than tears alone can fix. An ophthalmologist should be consulted in that case.
Systane or Refresh. Systane and Refresh are my favorite brands. Can't say when it will take effect because it depends on how bad your corneas are and if there are any other issues.