Nate, which currently has 85 kms per hour winds, is forecast to reach hurricane strength by the time it makes landfall in the United States late tomorrow on the north coast of the Gulf of Mexico.
New Orleans, where levees were breached during Hurricane Katrina in 2005, and other cities on the US Gulf coast were under hurricane watch.
The US National Hurricane Centre warned of possible "hurricane conditions" by tonight on Mexico's Yucatan Peninsula, where Cancun and other Caribbean resorts are located.
"Nate is expected to make landfall... as a hurricane," the Centre said.
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As of early today, the storm was located directly offshore from Belize, just south of the Yucatan Peninsula, with winds of 85 kms per hour that were expected to strengthen.
"The main threat to Belize is expected to be mostly thunderstorms and heavy rain which are causing localized flooding," the country's government said in a statement. It warned people in low-lying areas to go to higher ground, and for boats not to venture to sea.
Nicaragua bore 11 of the deaths, according to Vice President Rosario Murillo.
In Costa Rica, where a national emergency was declared, eight people died, including a three-year-old girl, after they were hit by falling trees and mudslides. An alert was issued for people to be wary of crocodiles that might be roaming after rivers and estuaries flooded.
Three other people were killed in Honduras.
More than 30 people are listed as missing in the three countries.
More than 5,000 people were put up in shelters in Costa Rica after having to abandon their homes because of flooding and the risk of unstable ground giving way.
In the Gulf of Mexico, some offshore oil and gas rigs were evacuated ahead of the storm's advance, the US government Bureau of Safety and Environmental Enforcement said in a statement.
The United States is recovering from two major hurricanes: Hurricane Harvey that tore through Texas in August, and Hurricane Irma in September.
Central America, the Caribbean, Mexico and the southern United States suffer an Atlantic hurricane season every year that runs from June to November.
The unstable weather brings heightened risk of flooding and mudslides in many poor Central American nations.
This year's season has been intense, with some areas in Central America getting up to 50 per cent more rain than average for September and October.
Costa Rica declared three days of mourning for those killed by Tropical Storm Nate, and President Luis Guillermo Solis, warned that, although the storm had passed, it was too early to say the danger had gone.
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