Gasoline, which cost about $1.50 here just last October, goes up this week to $2.25 a gallon. It is expected to keep climbing to nearly $3.00 a gallon before the end of the year, despite Britain's production of oil from its vast new North Sea fields.

Cutbacks in supplies by most of the major oil companies have made it difficult to buy gasoline at any price late at night or on weekends in many places. Fears of being stranded have caused many families to cancel plans for motoring vacations to more remote parts of southwestern England, Wales, northern Scotland and Ireland.

A more severe shortage of diesel fuel has prompted British Rail to make cuts in its passenger train service, forced some small trucking firms to buy fuel at exorbitant prices on a kind of black market, and left many farmers with little or no fuel to run their tractors.

Although gasoline prices here are still lower than in many parts of Europe and there are no long lines as yet at service stations like those in the United States or nearby for scarcer supplies of oil.

The supply situation aside, the price phenomenon is also explained by oil company increases and by conscious government tax policy. The budget introduced last week by the new Conservative government sharply increases taxes on oil products both as a way to raise revenues to offset tax cuts and to discourage consumption.

What has made the whole stiuation hard to take, and has caused a good deal of political controversy in recent weeks, is the paradox of an oil shortage suffered by a country about to become "self-sufficient" in oil from the North Sea.

The amount of oil being pumped from Britain's sector of the vast North Sea oil and gas fields now roughly equals 75 percent of the nation's consumption of oil. By some time next year, the North Sea harvest is expected to exceed Britain's needs.

After being prepared by proud politicians for the imminent arrival of "self-sufficiency," the British public is understandably surprised and frustrated to be facing an oil shortage instead.

The answer is that Britain exports more than half its North Sea oil to other countries. Energy experts and government officials knew that, but much of the public only now is becoming aware of it.

North Sea oil is such high-grade crude that British refineries could never run on it exclusively to produce the necessary broad range of oil products, although some retooling of refineries could enable them to use more North Sea crude than they can now.

The high-quality North Sea crude, however, brings top dollar on the world market. Its price is pegged with oil from Libya, Algeria and Nigeria, well above the official price for crude set by the Organization of Petroleum Exporting Countries. This means that for each barrel of exported, North Sea oil brings in more money than it costs to buy a barrel of OPEC crude to take its place in British refineries.

Exporting North Sea oil is thus more profitable for both the oil companies and Britain, which collects a large share of the profit in oil taxes and takes in more foreign currency to help its balance of payments.

Even if Britain were able to run its refineries exclusively on North Sea oil and has less to gain financially from exporting so much of it, there would be legal and diplomatic problems in hogging its North Sea oil. Britain and the oil companies operating in the North Sea have international obligations to share oil supplies and keep worldwide shortages spread out as evenly as possible.

This is what the new Conservative government's energy secretary, David Howell, has been telling Britain the past two weeks. He asked Britons to help cut the country's oil consumption by at least 5 percent, along with other Western nations.

"The world now faces a serious oil shortage which is unlikely to go away and will create continuing problems," he said. "It is not a passing phase but a permanent challenge. Nor does North Sea oil offer an escape route for Britain."

Nevertheless, under pressure from members of Parliament on all sides, reflecting their constituents' anger and impatience, Howell also has begun taking some steps and studying others to keep a larger proportion of the North Sea oil in Britain. These steps would include retooling some refineries, requesting that the oil companies redirect more oil to British customers as foreign contracts lapse, and collecting some of the government's North Sea royalties in oil rather than money.

Howell also is studying what to do with the British National Oil Corp., started by the previous Labor government and given certain competitive advantages over private companies in the North Sea, including an exemption from paying the oil tax. Prime Minister Margaret Thatcher has already decided to revoke the oil-tax exemption and Howell is likely to recommend that it also lose other special advantages as part of the Conservatives' drive to desocialize the economy.

Ironically, for complicated reasons, British National and British Petroleum, which is half owned by the British government, are the largest exporters of Britain's North Sea oil. The Thatcher government is expected to sell a large portion of its British Petroleum shares to raise money to help finance income tax cuts and decrease government involvement in private business, even though British Petroleum has become an increasingly profitable investment.