Why two British 5-inch gun projects were not direct successors

Was the later British 5-inch/50 gun the direct successor to the earlier 5.1-inch/50 project?

Not really; despite similar caliber and length labels, they were substantially different guns from different design generations.

The earlier 5.1-inch/50 project and the later 5-inch/50 weapon do not form a simple lineage. Their similar calibre and barrel-length labels make the comparison tempting. The underlying barrel systems, mechanisms, and design origins differ substantially. They belong to different generations of gun development, so a near-identical numerical designation does not establish direct succession.