When serving onboard ships I have heard it called both pilot house and bridge. Never thought about what the correct term was. I do remember a story about a seaman that decided to plop down in the captains chair......
The Booklet of General Plans for USS Miami CL-89 calls it the Pilot House.
See Plans No. 4 and 6. The entire level of the superstructure on which the Pilot hose is located is called the Navigation Bridge, with the Flag Bridge below. The plans show a configuration with an open walkway around the Pilot House. The photo in this thread shows an alternative configuration, with the walkway in front removed.
The pilot house is a named compartment in the navigation bridge in these plans. Normally a navigation bridge is a combined structure, sometimes several contiguous non-watertight compartments containing the conning gear (helm, EOT, binnacles), the chart table, radar PPIs, sometimes a small comm room, bridge wings which hold the peloruses, and other, such as the Captain's sea cabin. The navigation bridge is often one of several bridges on larger warships, including the flag bridge (where the admiral, if any, keeps an eye on the CO), and the signal bridge, where the flag hoists reside.
However . . .
in Plan 5, the title of which is "Bridges", the space in question is not called the pilot house but the _____ Bridge. It is the same space because it shows port and starboard gyrorepeaters, which are in the same space as the wheel. The word above "Bridge" is smeared and faded. I make it out as either "Open" or "Main", with Open more likely.