In Warwickshire (unlike some other areas which still have grammar schools) there isn't really such a thing as "passing the 11+". The schools have a fixed number of places and they fill them with students with the highest scores first - so that a score which gets a plae in one year, may not in another. In practice the scores required don't very enormously (although this year some of the schools have extra places, so the scores will probably be a bit lower for those schools than they have been historically).
From North Warwickshire you will probably be outside of the priority area for Lawrence Sheriff - although a few place, e.g. Ansty, Withybrook, do fall inside the outer priority area (check the priority circle map
). Places will be initially offered to those who live within the priority area. There are actually two parts to this - in inner and an outer area, which affects the scores required - typically those in the outer area require a higher score to get a place. If you outside of this circle then places cannot be offered in the first round, but may be offered from the waiting list to those with high scores if any of those initially offered a place subsequently turn it down (e.g if they opt for independent schools instead). In practice to get an out of priority area place you will need to be gettign a higher score than you would if you were inside the priority area.
From North Warwickshire, depending upon where you are and what the transport links are like, you could also have a look at the Birmingham grammars.