I dragged mine out of a field. No log book and coudn't read the chassis number either. There is a form the post office can give you, which I filled in with a letter explaining how I purchased the car and the fact that after 41 years the chassis number was unreadable, and six weeks later the log book turned up!
I think they write to the previous registered owner who verifies your story and then they OK it. Costs about £21 I think.