I can hardly express how appalled I am by your story. You have very strong grounds for complaint against both the school and also the county council as they are responsible for what is published in the "Going to School in Lincolnshire " book and so are also guilty of misleading you. So I would be inclined to write both to the Governors of the school and to the Director of Children's Services at LCC. If you get no satisfactory reply I would seek further advice from the Advisory Centre for Education and perhaps consider contacting your MP. (MP's enquiries do ensure someone senior has to look at your case).
I would have thought that you have a case for launching an appeal, but the success of any appeal will depend on other things. The appeal would be 2 stage - first against the failure to reach the required standard and second against the non-allocation of a place. You would need to show that he should have passed the exam had he had the proper notice (which I think would probably be considered to be long enough to have completed the NFER familiarisation papers). He will have needed to pass the VR exam and you will probably need other evidence of his high academic ability from school. Next you would need to show that he would have got a place had he passed along with all the others and his application been considered then. This is where you would need to know the numbers of places allocated to out of area applicants etc. Failing this you would have to show that it would be more prejudicial to your child not to have a place than it would be for the school to take in an extra pupil.
The more I think about this the more I think you may need some expert advice!! Try ACE - they would point you in the right direction even if they cannot help themselves. Web address is: www.ace-ed.org.uk
Wish you all the best whatever you decide to do.