Millions of people around the world are likely to be pushed back into poverty because climate change is undermining economic development in poor countries, the World Bank has warned.

Droughts, floods, heatwaves, sea-level rises and fiercer storms are likely to accompany increasing global warming and will cause severe hardship in areas that are already poor or were emerging from poverty, the bank said in a report.

Food shortages will be among the first consequences within just two decades, along with damage to cities from fiercer storms and migration as people try to escape the effects.

In sub-Saharan Africa, increasing droughts and excessive heat are likely to mean that within about 20 years the staple crop maize will no longer thrive in about 40% of current farmland. In other parts of the region rising temperatures will kill or degrade swaths of the savanna used to graze livestock, according to the report, Turn down the heat: climate extremes, regional impacts and the case for resilience.

In south-east Asia, events such as the devastating floods in Pakistan in 2010, which affected 20 million people, could become commonplace, while changes to the monsoon could bring severe hardship to Indian farmers.

Warming of at least 2C (36F) – regarded by scientists as the limit of safety beyond which changes to the climate are likely to become catastrophic and irreversible – is all but inevitable on current levels, and the efforts of governments are limited to trying to prevent temperature rises passing over this threshold. But many parts of the world are already experiencing severe challenges as a result of climate change, according to the World Bank, and this will intensify as temperatures rise.

Jim Yong Kim, the bank's president, warned that climate change should not be seen as a future problem that could be put off: "The scientists tell us that if the world warms by 2C – warming which may be reached in 20 to 30 years – that will cause widespread food shortages, unprecedented heatwaves, and more intense cyclones.

"In the near-term, climate change – which is already unfolding – could batter the slums even more and greatly harm the lives and hopes of individuals and families who have had little hand in raising the Earth's temperature."

The development bank is stepping up its funding for countries to adapt to the effects of climate change, and is calling for rich countries to make greater efforts at cutting greenhouse gas emissions.

Rachel Kyte, vice president of the World Bank, said it had doubled its aid for adaptation from $2.3bn (£1.47bn) in 2011 to $4.6bn last year, and called for a further doubling. She said the bank was working to tie its disaster aid and climate change adaptation funding closer together.

Aid from the bank to help poor countries cut their greenhouse gas emissions and pursue environmentally sustainable economic development stands at about $7bn a year, and is backed by about $20bn from regional development banks and other partners.

The report's authors used the latest climate science to examine the likely effects of global warming of 2C to 4C on agriculture, water resources, coastal ecosystems and fisheries, and cities, across sub-Saharan Africa, south and south-east Asia.

Kyte said the effects would be to magnify the problems that developing regions experience. More people would be pushed into slums, with an increased risk of disease. "We are looking at major new initiatives [in] cities; cities need billions of investment in infrastructure, but many developing cities are not really creditworthy," she said.

She pointed to Jakarta, where rising sea levels and decades of pumping freshwater from underground sources beneath and around the city were increasing its vulnerability to flooding. Choices would need to be made soon in many cities on how to stem the likely effects, but Kyte warned that the plans must be future-proof, citing Ho Chi Minh City in Vietnam, which has been forced to rethink its flood preparations despite spending $2bn on them.

Green campaigners emphasised the need to try to avoid 2C of warming, which scientists stay is possible if countries bolster their ambitions to cut greenhouse gas ambitions in the near future.

Stephanie Tunmore, climate campaigner at Greenpeace International, said: "Fossil fuels are being extracted in burned in the name of development and prosperity, but what they are delivering is the opposite.

"Some major impacts from climate change are already unavoidable and rich countries must urgently support the poor and vulnerable to adapt. But massive increases in the future costs of adaptation and damage can only be avoided by investing in a clean energy future now."

The World Bank has come under fire in the past for funding coal-fired power plants in some developing countries. However, it said the move was the result of old policies and was being phased out.