Captain Bligh (or Lieutenant as was his actual Naval rank at the time of the mutiny) did indeed discover what happened to the mutineers.
After he managed to successfully guide the longboat in which he and 18 loyal officers and crewmen had been set adrift 4000 miles to Timor in the Dutch East Indies, he returned to Britain where he was put under court martial for the loss of his ship. But due to what was described by the Naval Board as "a remarkable feat of seamanship" in getting the longboat to Timor with barely a chart and no compass, he was acquitted and promoted to Captain.
Following Bligh's report, the Roval Navy was intent on capturing the mutineers and bringing them to justice. The frigate HMS Pandora was dispatched under the command of Captain Edward Edwards. The Pandora arrived in Tahiti on March 23rd 1791, and was met by three Bounty crew members who informed Edwards what had happened following the mutiny.
Once arriving back at Tahiti, conflict broke out almost immediately amongst the mutineers, leading them to split into two groups. Sixteen men decided to stay on Tahiti while Fletcher Christian and the remaining mutineers, plus eight Tahitian men and twelve women, fled to find a safe haven from the Royal Navy somewhere in the South Pacific. Of the sixteen that remained on Tahiti, two were murdered by other mutineers and the rest were captured by Captain Edwards, who intended to return them all to Britain for a court martial. Four, however, died along the way after the Pandora ran aground in the Great Barrier Reef off the coast of Australia.
Bligh, now seen as something of a hero, returned to Tahiti himself several months after the Pandora on HMS Providence alongside HMS Assistance, to complete the task he had been assigned with the Bounty. However, upon his return to Britain he discovered Captain Edwards had returned the ten captured mutineers who had been put on trial. Bligh's reputation had been tarnished, much to his anger and outrage, as the men testified to the ill treatment they received under Bligh's command. What infuriated Bligh further is that four of the men were acquitted of the crime of mutiny; three were convicted, but either received a royal pardon or were otherwise excused; and only the remaining three were found guilty and executed for their crimes.
To make matters worse, not only did it appear to Bligh that his former friend and ringleader of the mutiny, Fletcher Christian, had escaped justice, the public were now sympathising with him. On top of which he was put on half pay and unemployed for 18 months. Bligh returned to active service in 1795 and rescued his reputation by serving with distinction under Admiral Duncan at the Battle of Camperdown in 1797 and with Nelson at the Battle of Copenhagen in 1801.
Bligh finally discovered what happened to Fletcher Christian in 1810, after he returned to Britain following four years as Governor of New South Wales in Australia and a promotion to Rear Admiral.
In 1808 when the American seal-hunting ship Topaz encountered Pitcairn Island. On the island, the crew of the Topaz found a man named Alexander Smith (also known as John Adams), who claimed to be the last surviving member of the original crew of mutineers on the Bounty. Christian and the crew had stumbled upon Pitcairn, then an uncharted island, and elected to settle there, destroying the Bounty to reduce the chances that they would be identified with the mutiny. The crew found no paradise on Pitcairn, however. Conflict between the Bounty mutineers and the Tahitian men crew violent, claiming the lives of all of the crew, including Christian, who died of a gunshot wound, except four. One of the men became intoxicated and fell off a cliff to his death. Another was killed by the two remaining mutineers, after he attacked them. The sole remaining crew member aside from Smith succumbed to natural causes.
William Bligh died in London in 1817. But in April 2015, on the anniversary of the mutiny, the feud between Bligh and Christian was finally put to rest when 70 year old Maurice Bligh, the great-great-great-grandson of the famous captain, and 40 year old Jacqui Christian the great-great-great-great granddaughter of the lead mutineer who lives on Pitcairn, greeted each other in Tahiti. Christian brought the Bible that her mutineer relative stole from the Bounty's captain in a gesture of goodwill. Bligh, in an act of friendship, then returned the Bible to the resident of Pitcairn.