Rescue officials tried to send a second helicopter to the site but it was unable to fly due to strong winds, police spokesman Sailesh Thapa told AFP.
No climber has reached its summit for more than 20 years.
Helicopter pilot Siddartha Gurung was among the first people to reach the site after the deadly storm and described a scene of total destruction with the tents flattened and the climber's bodies scattered across the area.
The bodies of four South Korean climbers, two Nepalese guides and another unidentified person was discovered by villagers that trekked to the base camp. The other South Korean climbers who died have been identified as Lee Jaehun, Rim Il-jin, Yoo Youngjik, and Jeong Joon-mo. The mountain Mount Gurja, with a 7,193-metre peak, is in western Nepal.
"The camp was completely destroyed", Myagdi district official Liladhar Adhikari was heard saying, according to the BBC.
Sherpa added that an alarm was raised after 24 hours elapsed without contact from the group.
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Public broadcaster TVE showed footage of flooded streets and overturned cars along the guardrails of a major road. The two British citizens were travelling in a taxi when a river burst its banks on the popular holiday island.
The climbers had been waiting for the bad weather to clear so they could begin their climb on the unsafe Himalayan peak. A police team was heading toward the base camp on foot and will likely reach there on Sunday, Budamagar said. "Going by the color of their skin four bodies must be of Korean climbers, while two are Nepalis", police official Bir Bahadur Budhamagar told Reuters.
It was the deadliest Himalayan accident since 19 people were killed at the base camp of Everest in 2015, after an avalanche set off by a devastating quake.
The previous year, 16 Sherpas were killed on Everest when an avalanche swept through the Khumbu Icefall.
The rarely-climbed Mount Gurja sits in Nepal's Annapurna region, next to avalanche-prone Dhaulagiri, the world's seventh highest mountain.
Nepal's main climbing season runs between April and May, when weather conditions are favourable. No-one has stood on it's summit since 1996.