You can read Lincolnshire's Fair Access Protocol here:
http://www.lincolnshire.gov.uk/upload/p ... tocols.pdf
However, you will see that a pupil arriving in-county in-year would have to be deemed "hard to place" for the local authority to direct admission and that would mean that neither the parent nor the LA could find a suitable school place within reasonable distance of the pupil's home. As Lincolnshire have been prepared to send secondary school age children 30 miles away before now and there are likely to be places available in schools far nearer to Bourne it would be highly unlikely that your child would be considered hard to place. The protocol is very much a means of finding a
place not a means of getting a child into a particular school.
When you appeal for a Community or Controlled school it is usual for the LA to say what school place it would be able to offer the child and whether free transport would be available.
It is normal for the LA to require some proof of residence before allocating a school place or setting up an appeal. For Crown Servants a posting notice is sufficient - but if this does not tie you in sufficiently to Bourne itself you may need to wait until exchange of contracts on the house.
Very generally speaking appeals to Grammar schools in Lincolnshire for children who live very near to the school (where it is a Community/Controlled school) and where the child has passed the tests have a fairly good chance of success. You will need to present all your relevant arguments for an oversubscription appeal.