Why a British 130 mm gun would probably stay 130 mm

How likely was Britain to turn a proposed 130 mm gun into a 127 mm weapon?

Unlikely; 130 mm was chosen for cruiser firepower, and reducing it to 127 mm would weaken the purpose of the program without a new requirement.

A program begun around 130 mm would have selected that caliber for a reason, primarily to provide cruiser-level firepower below the traditional 6-inch gun. Reducing it to 127 mm would move the weapon toward the destroyer-oriented 5-inch category. The Royal Navy had already required persuasion to accept a cruiser gun below 6 inches, so a further reduction would cut against the original goal. Unless the procurement requirement changed, the awkward middle position of 130 mm would be retained rather than rounded down for nominal commonality.