The first named storm of the Atlantic hurricane season is posing an uncertain threat to the Gulf of Mexico, even as efforts to contain the worst oil spill in U.S. history are set to ramp up.

For now, Tropical Storm Alex, which is hitting the western Caribbean with rain and high winds, is not expected to pass close to BP Plc.'s blown-out well off the Louisiana coast.

But even a miss that only generates large waves could greatly complicate clean-up efforts from Louisiana to Florida

Current official estimates suggest between 35,000 to 60,000 barrels a day are leaking from the rogue well. BP collected over 24,000 barrels on Friday and about 11,640 barrels in the first half of Saturday, the company estimated.

New equipment being moved to the site of the leak in the coming week could raise the daily collection rate to 53,000 barrels a day, U.S. Coast Guard Admiral Thad Allen, who is coordinating the U.S. oil spill response, said on Saturday.

U.S. Energy Secretary Steven Chu and Interior Secretary Kenneth Salazar are also scheduled to review plans on Wednesday for a new containment system that could boost collection to 80,000 barrels per day, Allen said.

The ultimate solution to plugging the undersea gusher still lies in a relief well being drilled by BP. On Friday, the company said drilling is on track to intercept the blown well.

But progress could be scuttled if Alex, or a subsequent storm, comes too close to the leak site.

Allen said BP would be forced to evacuate the vessels and personnel working to contain the spill if a storm with gale-force winds, 39 miles per hour or stronger, were expected within five days at the leak site. That would once again leave the well gushing uncontrollably.

For now, Alex does not pose such a risk -- a rare piece of good news for the unprecedented oil spill disaster now in its 69th day. We understand it's moving westerly at this time and does not threaten the site, said Allen.

Late on Saturday, Alex made landfall in Belize and dumped showers on northern Guatemala and the Yucatan Peninsula. The storm had sustained winds of near 40 miles per hour (65 km per hour) and was located about 55 miles southwest of Chetumal, Mexico, cording to the U.S. National Hurricane Center. The storm was expected to slow as it moved over Belize and the Yucatan Peninsula overnight, the center said.

But when Alex moves back out over the warm waters of the southern Gulf of Mexico, the storm is expected to strengthen again.

In Grand Isle, Louisiana, a tiny village jutting out into the Gulf of Mexico and a haven for commercial and recreational fishing, residents anxiously awaited updates on the storm.

If it comes and it's somewhat severe, you might as well say goodbye to Grand Isle, said Pam Brooks, 50. The oil will get thrown up and coat everything.

The Gulf disaster and its impact on London-based energy giant BP was on the agenda on Saturday when U.S. President Barack Obama and new British Prime Minister David Cameron held their first face-to-face meeting.

Shares of BP, a staple holding of many U.K. pension funds, have been savaged since the oil crisis started and fell another 6 percent to a 14-year low on Friday.

Investors are fretting about the potential costs to BP, which include but are not limited to a $20 billion compensation fund it set up under intense U.S. political pressure. BP said it has paid out $2.35 billion so far in clean-up and compensation costs related to the spill.

Obama has been highly critical of the company, even as his own poll ratings have fallen amid perceptions that his handling of the Gulf crisis has been too slow.

Cameron and Obama agreed that there was nothing to be gained from damaging BP as a going concern, a UK official said after the leaders met at the G8/G20 summit in Canada.

The pair also agreed that BP must meet its obligations to cap the leak, clean up the damage and meet legitimate compensation costs.

LEGAL WRANGLING, PROTESTS

The Obama administration on Friday asked a U.S. appeals court to stay a ruling from a federal judge that overturned a six-month ban on new deepwater drilling in the Gulf.

On Saturday, the state of Louisiana filed a brief opposing the administration's request. Each day the ban is in place, millions of dollars of income are lost to the citizens of Louisiana, and by the state, the brief said.

The disaster is also taking a heavy toll on fishing, tourism and the environment. About one-third of U.S. federal waters in the Gulf of Mexico remain closed to fishing.

The U.S. Fish and Wildlife Service reports a growing toll of birds, sea turtles and marine mammals, mostly dolphins, found dead or debilitated along the Gulf Coast.

In Florida on Saturday, hundreds of local residents and supporters protested the spill with a 15-minute Hands Across the Sand demonstration calling for a halt to U.S. oil drilling in the Gulf.

It a sick feeling, said Cindy Nevens, resident of Navarre Beach, Florida. There is no end in sight and we don't know if it can be stopped or if they are telling us the truth about how much oil they are collecting.

(Additional reporting by Erwin Seba in Houston, Sumeet Desai in Toronto, Jose Cortazar in Cancun, Cyntia Barrera Diaz in Mexico City, Steve Gorman in Los Angeles, Jeremy Pelofsky in Washington, and Mike Peltier in Florida; Writing by Ros Krasny; editing by Todd Eastham)