Yamato would probably be scrapped after the war. Giving it to somebody else is highly unlikely. The Nagato would be used in the nuclear tests - part of the reason it was used was because it had been Admiral Yamamoto's flagship during the attack on Pearl Harbor.
If we're talking interesting fantasy results, I'd say the best one is that it gets taken by the USN as a war prize, and sailed to California in order to do a major examination of it, as the Americans would be looking to examine it, just to see what it is capable of. It goes straight to Long Beach Naval Shipyard, where the US rip the thing apart to find out as much as they can about it. It spends most of 1946 and 1947 there as a result. Nobody in the USA is up to scrapping something so big right after the end of WWII, so the Yamato is moved to the reserve fleet at Suisun Bay and stays there through the 1940s and 1950s. A proposal is brought up to give the Yamato back to Japan - the massive battleship, while obsolete in modern warfare, would likely still be a major point of pride for Japan - or even to commission the brute into the USN to fight in Korea, but neither comes to pass. Japan becomes fully independent in 1952, and over the 1950s and 1960s Japan grows dramatically in terms of economics, and becomes a staunch ally of the United States.
In 1972, the same agreement which gives Okinawa back to Japan also sees Japan ask for the return of the Yamato. It's useless in modern warfare, but the Japanese envision it being turned into a museum ship. In August 1972, the Yamato is towed back to Japan and is docked in Nagasaki, and after a major cleanup and repair, is opened to the public in September 1974.
In the 1980s, Japanese politicians and their American counterparts came to the agreement that Japan was important to the United States. A faction of the Japanese Maritime Self-Defense Force took heart from Reagan's decision to recommission the Iowa class battleships and in 1982, Mitsubishi Heavy Industries and this faction proposes bringing Yamato back to life as a warship, while also salvaging its sunken sister ship, Musashi, for use as the museum. This movement gains massive traction in Japan, and is appealing to many Japanese politicians - the battleship would be a very suitable flagship for the Japanese Naval Forces, while not likely being too antagonizing to its neighbors. The Americans raise no real objections to the idea, though privately they are somewhat surprised that the Japanese would have the chutzpah to bring back one of its most fearsome weapons of World War II. The idea is approved in August 1983, and the ship is towed to Mitsubishi's shipyard in Nagasaki and begins modernization.
The modernization plans for the Yamato are considerably more extensive than those done on America's Iowa class battleships. The AA guns and 155mm are removed in their entirety, as are two of the 5" mounts. The insides of the main gun turrets, which had been largely dismantled by the Americans and never repaired, were gutted and rebuilt using two-stage hoists and adding newer powder bags. Like the Iowas, Yamato is fitted with Harpoon cruise missiles, though the ship is also fitted with Sea Sparrow anti-aircraft missiles, which are fitted in pop-up assemblies allowing them to not be damaged by the immense gun backblast. The exhaust stack and tall "pagoda" are dismantled and replaced, and the new superstructure provides far more space, removing any potential problems with space requirements.
The ship's boilers are replaced with electronically-controlled units, which in addition to massively reducing the number of crewmen needed to operate the ship, improved the ship's power from 150,000 shp to 220,000 shp, improving the ship's top speed from 27 knots to 30.5 knots, improving the ship's fuel economy and massively increasing the ship's electrical generating capacity. The 155mm secondaries got new guns and hoists, improving their firing speeds, and the ship was fitted with four Phalanx CIWS units. The aircraft and boat launching facilities are removed, and the rear of the vessel is reconfigured to allow the gaps used by these facilities to be used as a helicopter deck and hangar, used by two SH-60J or HSS-2B helicopters. In addition, the ship was fitted with far-improved creature comforts, including air conditioning and the latest in modern electronics and communication systems, including a 3D radar system, modern fire control and electronic warfare systems and many other systems improvements.
Fitted as such, JDS Yamato commissioned into the JMSDF with some fanfare on April 15, 1986. Her first journey, fittingly enough, was to sail to San Francisco, where she was met the newly-recommissioned USS Missouri in San Francisco. There, Yamato met her WWII nemesis for the first time, and many wondered which of the two would have been victorious had they taken each other on - but most of the crews of both vessels agreed that it would have been a tough fight for both vessels, and that it was a good thing that they were now on the same side.
Yamato spent much of 1986 and 1987 sailing around the Pacific Ocean, visiting ports around the Pacific. While she was warmly welcomed in several ports - including Sydney, Singapore, Manila, Vancouver and a number of American ports, including San Francisco, San Diego, Los Angeles and Seattle - she was not allowed to visit China under any circumstances, and she was only allowed to dock in Korea just before the 1988 Olympics in Seoul. Yamato made an appearance at the RIMPAC exercises in 1987, doing a gunfire demonstration for the assembled vessels.
In February 1988, Yamato left Yokosuka for the first around-the-world cruise by vessels of the Japanese Maritime Self-Defense Force. First stopping in Vancouver, she made her way down the west coast of the Americas, stopping San Francisco, Los Angeles, Acapulco, Lima and Valparaiso, before sailing around Cape Horn. Calling again at Buenos Aires and Rio de Janiero, she skirted the Caribbean, calling at Miami and New York in late April. Crossing the Atlantic, the ship made the first ever visit by a Japanese naval vessel to Great Britain (at Portsmouth), as well as calling at Rotterdam, Wilhelmshaven, Lisbon, Barcelona, Toulon, Rome and Athens before sailing down the Suez Canal. Calling again at Bombay, the ship sailed to Perth, Australia, via Colombo in Sri Lanka. Visits to Melbourne, Auckland, the Fiji Islands and Brisbane followed, before the ship steamed north back to Japan via Guam and Okinawa, arriving in Kure on August 15, 1988. The ship was there but a month before she sailed to Incheon, South Korea, for the opening of the 1988 Summer Olympics.
JDS Yamato remains in the Japanese Navy as its flagship, having taken the mantle of the world's last battleship when USS Missouri was decommissioned on March 31, 1992. The ship was the first ship to be renamed JS Yamato after the Ministry of Defense was upgraded to a full ministry in 2007. The ship's high operating costs meant that it did not sail as much in the 1990s and early 2000s as it had in times past, but the ship is still able to act as a full warship, and owing to her almost-mythical status among the Japanese, is likely to remain so for some time to come.
OOC: I know this has giant holes. But I thought it cool - and Yamato
does have something of a legend in Japan.
