The British government's plan for offering private investors a majority share in Britoil, the oil-exploration and production arm of the state-owned British National Oil Corp., fell far short of expectations today, with buyers picking up only a third of the stock issue.

The rest of the 255 million shares, underwritten by major banks and financial institutions, will begin trading next week on the London Stock Exchange, but probably at a figure well below today's asking price to subscribers of 2.15 per share (about $3.50).

The public's coolness to the offering is a setback to Prime Minister Margaret Thatcher's hopes for returning other major nationalized companies -- including British Telecom, British Airways, the British Steel Corp. and auto manufacturers BL Ltd. -- to private ownership. Only British Telecom, the rough equivalent of AT&T, is now profitable, although British Airways reports that it is struggling out of the red.

But Britoil, carved out of BNOC last summer for the specific purpose of offering it to investors, seemed a particularly good bet for a successful de-nationalization program. The company has a stake in almost one-third of all North Sea petroleum assets and its estimated market value after the sale will be just short of $2 billion.

By turning the shares over to leading British financial institutions for underwriting, the government itself faces no risks on the transaction, although it will definitely make less on the sale than it had hoped.

However, for the six merchant banks and five stockbrokerages who are the underwriters, failure to quickly dispose of the vast amount of Britoil stock that remains after today is likely to make them less disposed to future government de-nationalization efforts.

A number of factors were blamed for the poor response to the Britoil shares. One is the generally soft oil market, which yesterday prompted Saudi Arabia's Oil Minister Sheikh Yamani to say the Saudis are considering a cut in the $34 per barrel price for oil at next month's meeting of the Organization of Petroleum Exporting Countries.

Britain's Press Association quoted economist David Morrison of Simon and Coates brokers as saying: "Everybody is extremely depressed about the prospects for oil next year. There is no pickup in world trade in prospect and OPEC prices are in disarray . . .

"The government should not have gone ahead with the issue now. At a different period it might have been wonderful, but the environment is not right," he said.

Other financial sources said that Britoil had suffered from some adverse publicity about its financial structure, management, proven reserves and long-term prospects. Moreover, the opposition Labor Party has said it would re-nationalize the company if it returns to power at the next election.

There is also thought to be uncertainty among investors caused by the sharp drop in the value of the British pound in recent days, although this is actually a plus for Britoil because oil prices are set in U.S. dollars.