The torrential downpours across the densely populated region are critical to replenishing rivers and groundwater, but also cause widespread death and destruction.
In India's poorest state Bihar, home to 125 million people, 11 have died and at least four million have been affected by heavy rains, with homes and villages damaged.
More than 300,000 villagers were evacuated to relief camps and officials warned of further cloudbursts in the next two days.
The eastern state, which has a fragile healthcare system and has recorded more than 46,000 COVID-19 infections, entered a fresh two-week lockdown in mid-July as virus cases spiked.
Officials fear the flooding could worsen the spread of the infectious disease.
"Reaching out to flood victims and providing them help is not easy due to fear of the pandemic," relief worker Mahendar Yadav said.
In Assam, where at least 50 people have died in the past 10 days, floodwaters were receding as officials scrambled to help 1.5 million people whose homes and communities were damaged.
More than 37,000 villagers remain in relief camps and special attention was being given to health-related issues, Assam Disaster Management Authority coordinator Pankaj Chakrarvarty told AFP.
"It's not always possible to maintain social distancing in the camps," said one evacuee, Ranjit Rabha.
"Thankfully, here in our camp, we don't have any COVID-19 positive residents."
At Assam's World Heritage-listed Kaziranga National Park, 143 animals have died so far, including 16 rare one-horned rhinos.
'Allah will decide'
India's superstar cricket captain Virat Kolhi and Bollywood actress wife Anushka Sharma on Thursday pledged to support flood relief in Bihar and Assam.
Bollywood star Priyanka Chopra Jonas, who was born in Bihar, also promised to donate to relief efforts.
In Bangladesh, which is criss-crossed by rivers, 129 people have died from the floods and flood-linked illnesses, officials said.
About one-third of the delta nation remains under water. The floods have dragged on for a month in what authorities said was the worst inundation since 2004.
"The house is underwater, no food, not enough drinking water," farmer Muzaffar Ali from northeastern Sunamganj district told AFP.
"In this situation who cares about social distancing or wearing masks? I leave it to Allah. He will decide."
Nepal has been badly hit by landslides and floods, with at least 155 people killed and 57 missing since mid-June, according to the home ministry.
"The threat of COVID-19 adds challenges for our rescue and search teams," National Emergency Operation Centre chief Murari Wasti told AFP.
The number of landslides has increased this year, with experts saying the massive 2015 earthquake and more road construction could be triggering the deadly slips.
"The earthquake... it loosens the earth and heavy monsoon rains can cause disturbance. This year could be a cumulative effect," geologist Basanta Raj Adhikari told AFP.