Why French 130 mm ammunition would not fit

Could British 4.7-inch or 5.25-inch guns use ammunition made for French 130 mm weapons?

No. The British 4.7-inch was about 120 mm and the 5.25-inch about 134 mm, while roughly 5.1 inches corresponds to 130 mm.

The proposed ammunition swap fails on dimensional grounds. A British 4.7-inch bore is approximately 120 mm, not 130 mm. The 5.25-inch weapon is larger at roughly 134 mm, while a nominal 5.1-inch gun lies closest to 130 mm. Similar labels and neighboring calibres therefore do not make shells or cartridges interchangeable.