The Swiss-based organisations said there had been a 95 percent reduction in passenger flights, which are typically used to transport mail.
Meanwhile there had been a 25-30 percent increase in demand for online purchasing during lockdown restrictions aimed at curbing the spread of COVID-19.
Postal administrations are therefore facing a challenge in shipping international mail, the statement said.
"The cancellation of more than 4.5 million passenger flights - the primary means of transporting post - has meant that capacity is scarce, costs more and takes longer," said UPU director general Bishar A. Hussein.
"Action needs to be swiftly taken to address the shortfall in air cargo capacity and to keep the mail moving."
The statement urged governments to remove border blockages, avoid unnecessary regulations and speed up issuing permits for chartered operations.
IATA said it was helping to get mail onto cargo flights by providing information on availability and new alternative routes.
"It's vital that everything is done to support the smooth movement of mail which is an important component of society," said IATA director general Alexandre de Juniac.
The UPU is a United Nations agency which coordinates the global postal system, while IATA represents some 290 airlines comprising 82 percent of global air traffic.