The London Metal Exchange is in the throes of the worst crisis of its 108-year history.

It began 10 days ago with the collapse of the international tin cartel, which for 29 years had controlled world tin prices.

Quickly, the trouble in tin spilled over into other metal markets, creating the most serious turmoil since the silver market was sacked by the unsuccessful effort of the Hunt family to corner the market.

The tin crisis now threatens to bankrupt several LME brokers and to inflict deep, possibly permanent, damage on London's reputation as a financial center.

The situation is so serious to London's tin brokers that they are pleading for a British government rescue as the only solution.

"When you've got 22 countries, 28 LME member firms and dozens of banks all running around like headless chickens, then you need somebody to take a lead," said David Williamson, a broker at Shearson Lehman Brothers in London. "That has to be the government."

The crisis has been brewing for months, as the continuing oversupply of tin on world markets obliged Pieter De Koning, the buffer stock manager for the 22-nation International Tin Council, to buy up larger and larger quantities to keep prices within the agreed support levels. In September, De Koning, his funds nearly exhausted, asked the ITC's tin-producing nations for more cash so he could keep buying.

"They promised him 60 million British pounds ," said Peter Kettle of London Commodity Research Unit, "but none of it turned up. They just took too long."

On Oct. 24, what London metal brokers had feared for so long finally happened: The buffer stock manager ran out of money. Overnight, the price plunged from $5.43 a pound to $4.50 for tin that five years ago was selling for $8.50 a pound. At the request of the tin industry, tin trading was suspended by the LME and the Kuala Lumpur Tin Market.

Fear for the future of the London Metal Exchange caused the crisis to spread to the nickel, zinc and aluminum markets, and by the end of the week had big North American mining companies worried. Inco Ltd. of Canada, the biggest nickel producer in the noncommunist world, asked the LME to suspend trading in nickel after the price of that metal dropped to its lowest level in three years. Another Canadian mining company, Falconbridge Ltd., urged a halt in zinc trading, and Aluminum Co. of America announced it no longer would use LME trading in determining its prices.

All last week, the tin market remained in a state of suspended animation as brokers met with bankers, bankers talked with British government officials and government officials spoke to their foreign counterparts. On Friday, the LME announced it would not reopen the tin market next week as hoped for, but will keep it shut for at least another week. The same day, the British government announced that it would convene a meeting of the Tin Council next Wednesday.

"It's the first hopeful sign I've seen," Kettle said. "Next week's meeting of the Tin Council might just lead to a bailout."

Brokers fear that, without government intervention, the price of tin is certain to collapse, perhaps dropping as much as 50 percent. That would lead to bankruptcy for several London brokers, which are now owed hundreds of millions of pounds by the penniless tin cartel.

Beyond the failure of the tin brokers lie worries about the larger financial community.

"The ripple effect would extend to other metal traders, to banks and internationally," said Williamson, the Shearson Lehman broker. "It could destroy the LME, and it could destroy Britain's reputation as an orderly, well-regulated place in which to do business."

Some insiders claim that underneath statements about financial stability lay worries about revelations of foolish, irresponsible, and possibly even criminal, activities, as the ITC has struggled to keep the tin price afloat in the face of a grossly oversupplied market.

"The mess is much worse than it seems," said one broker, who asked not to be named. "The buffer stock manager almost certainly broke the rules. He's been pledging metal two or three times over, it seems. There have been all sorts of infringements of the rules and regulations."

Alarmed with these frightening scenarios, the LME and its members have been pleading for help from the British government. On Friday, Trade Secretary Leon Brittan said the government would be prepared to honor its obligations, as an ITC member, to tin traders. Metal traders said this looks like the start of the bailout the brokers want, but it remains to be seen next Wednesday whether the other member nations will take the same view.

Beyond the immediate crisis lies the question of the long-term future of the tin cartel. Founded in 1956, it is the oldest commodity agreement still functioning. In the view of many brokers and economists, the cartel has been hopelessly out of touch with reality for years, trying for political reasons to keep tin prices at unrealistic levels.

High tin prices have driven up production, while demand for tin has fallen because of the increased use of aluminum and synthetic substances in its place. To keep the price up, the ITC resorted to export controls on the producers that are members of the cartel.

This only encouraged nonmember countries to expand their tin production. From 80 percent of total worldwide production in the 1960s, the ITC's share has fallen to about 56 percent today. In the past five years, nonmember Brazil has quadrupled its market share to 13 percent. Next year it will be the second-largest producer in the world, after Malaysia.

When the ITC members sit down next week, they will have to choose one of three courses: They can come up with enough money for the ITC to pay its debts to the London brokers and try to keep the whole price-fixing system going. This would paper over losses, but store up trouble for the future. The British government is likely to oppose this. "It leads to an open-ended blank check," commented one official.

The ITC also could wash its hands of the whole affair, either consciously or by failing to agree on what to do. This would put the ball back into the British government's court, so far as the LME's problems are concerned. The Thatcher government then would have to decide whether government aid to the London Metals Exchange is necessary.

Lastly, the ITC might guarantee the ITC's current debts as part of a plan that would include gradually dismantling the price-fixing cartel for tin.

This is the solution most economists and brokers now favor. It would solve the LME's problems, but if forecasts of a 50 percent drop in the tin price are accurate, it would mean grave economic consequences for tin-producing nations. Malaysian officials have talked about closing all but 22 of the country's 450 mines. In Bolivia, which gets 40 percent of its export revenue from tin, the consequences would be disastrous.

However disastrous, most observers now feel a return to a free market in tin is unavoidable. That was the view of the Reagan administration, when, in what now looks like an astute move, it pulled out of the ITC in 1982.

One broker here summed the situation up this way: "The ITC is finished. We will see a free market in tin. The LME still has a future, perhaps with a few reforms in its operations. If there's one big moral in the story, it's 'never take a government's word for anything.'