Specifically with regard to the 11+, my own LA has the following policy:
Very occasionally, a child who is over the age of 11 will be accepted for the 11+ tests. This will be either due to educational reasons in the past which resulted in a decision to place your child in a lower year group and which have continued, or if a child has missed a year or more of schooling through illness or living abroad. Going to a school in another country is not, in itself, a reason for taking the 11+ at an older age. The tests should be sat, in this case, at the same time as the year group that the child has normally been working with. Again, before a decision is made, we would ask your child’s current headteacher for details surrounding the circumstances to place your child in a lower year group, this must not be just because your child was born late in the school year.
You will need to check with the relevant admission authority what their policy is.