Ukrainian forces, still holding positions in western Bachmut, have destroyed key bridges across the river and are able to shell Wagner group forces from fortified buildings.
This has resulted in the area around the river becoming "a death zone", which likely makes it difficult for the Wagner soldiers to continue advancing west, the department writes further.
However, the Ukrainian forces and their supply lines in the west remain vulnerable to attacks from the north and south, according to the British.
Wagner: Conquered Eastern Bachmut
The leader of the Wagner group, Yevgeny Prigozhin, has previously claimed that his forces have taken over the entire eastern part of the city.
"Wagner forces have taken over the entire eastern part of Bachmut, everything east of the Bachmutka River," Prigozhin said in an audio clip distributed by his press service via the Telegram messaging app on Wednesday.
Important for both sides
The battle for the town of Bakhmut in the Donetsk region of eastern Ukraine has been going on for six months and is the longest and bloodiest of the war so far.
It is described as decisive by both sides.
Russian Defense Minister Sergei Shoigu claims that a Russian conquest of the city is a prerequisite for the country's forces to be able to continue advancing westward in Ukraine.
Ukrainian President Volodymyr Zelenskyi, on the other hand, believes that a Russian conquest would be devastating as it would give Russia "an open road" to continue on.