KYIV, May 25 (Reuters) - Russia's Wagner private army started handing over its positions in Bakhmut to regular Russian troops on Thursday, five days after declaring full control of the devastated eastern Ukrainian city following the longest and bloodiest battle of the war.

Moscow says capturing Bakhmut opens the way to advances in the eastern industrial region known as the Donbas. Kyiv says the battle drew Russian forces into the city, inflicting high casualties and weakening Moscow's defensive line elsewhere.

Wagner founder Yevgeny Prighozin, who has repeatedly accused Russia's regular military of abandoning ground captured earlier by his men, said Wagner would be ready to return to the city if needed.

"From today at five in the morning, May 25 until June 1, most of the (Wagner) units will rebase to camps in the rear," Prighozin said in a video, wearing battle gear and standing beside a war-damaged residential block. He says 20,000 of his fighters died taking Bakhmut.

Ukrainian Deputy Defence Minister Hanna Maliar said in a message on the Telegram app that Wagner had handed over positions on the city's outskirts but "inside the city itself Wagner fighters remain".

Russia moved ahead on Thursday with a plan to deploy tactical nuclear weapons in its neighbour and close ally Belarus, signing a deal about the storage of the warheads. It would be the Kremlin's first deployment of such bombs outside Russia since the 1991 fall of the Soviet Union.

"The movement of the nuclear weapons has already begun," Belarus President Alexander Lukashenko told reporters in Moscow, where he was attending talks with other leaders of ex-Soviet states.

The U.S. State Department denounced the deployment plan, but said Washington had no intention of altering its position on strategic nuclear weapons nor had it seen any signs Russia was preparing to use a nuclear weapon.

"It's the latest example of irresponsible behaviour that we have seen from Russia," U.S. State Department spokesperson Matthew Miller said.

It made the same threat and demands in March. Moscow then agreed last week to renew for 60 days the Black Sea export pact - initially brokered by the United Nations and Turkey last July with Russia and Ukraine to try to ease a global good crisis aggravated by Moscow's February 2022 invasion.

Some peace initiatives have been proposed, including by Russia's ally China, the Vatican and a group of countries in Africa but there are no signs of actual talks to end the war.

"There is no force that can oblige Ukrainian society and its leaders to talk with the Russians. Not while Russian troops remain here. That just doesn't exist," Ukrainian President Volodymyr Zelenskiy's chief of Staff, Andriy Yermak, was quoted as saying on Thursday in an interview with Interfax Ukraine news agency.

Russian forces have taken over parts of Ukrainian territory since the war began and Ukraine is planning a major counteroffensive to retake land, using modern weapons supplied by Western allies and troops freshly trained across Europe, but details are scant.

Russia has built sprawling fortifications in eastern and southern Ukraine in readiness for the counteroffensive. Military analysts said small offensives carried out by Ukrainian forces may be aimed at spreading Russia's troops thinly, creating gaps elsewhere that can be exploited.

In a prisoner swap with Russia, Ukraine received 106 soldiers who had been captured in Bakhmut. Russia confirmed the swap, saying Wagner took part in the exchange, but did not say how many Russians were released.