Are we now talking about how Lewis regarded himself as opposed to what he actually was by birth and ancestry? Because those are entirely different questions.
As I said, it's fine to call him British because he belonged to one of the countries that made/makes up the British Isles (although as Lewis points out in 'The Four Loves' it is only foreigners who actually talk about 'Britain'.) However, each consituent part of what we call 'Britain' is also a nation in its own right and the nation that Lewis belonged to was Northern Ireland. Thus he was Northern Irish, Irish or British (if you like) but not English.
*goes away muttering about Jack Charlton and Craig Brown with their dubious recruitment policies for international football...*