UNDER the fierce pressure of the Thatcher government's militant conservatism, the opposition Labor Party is moving sharply leftward. It may also be breaking up. The comfortable cliches about British politics, and its tendency to draw power to the center, are being rapidly eroded.

James Callaghan, former prime minister and the party's leader, was repeatedly defeated by the doctrinaire left in the party's annual conference two weeks ago. Having had enough, Mr. Callaghan has now resigned the leadership. The immediate question of his successor is less interesting than the changes overtaking the character of politics in Britain.

Over Mr. Callaghan's vehement objections, a narrow majority at the conference committed the Labor Party to pull Britain out of the Common Market. The same narrow majority pledged the party to much wider nationalization of industry, and to unilateral nuclear disarmament. Within the minority -- Labor's right wing, although well to the left by American standards -- there was increasingly serious talk about pulling out and setting up a new social democratic party.

Mr. Callaghan lost control of the conference because of the rapid swing leftward by some, although not all, of the unions. Traditionally a force for political moderation, the unions are now as badly divided among themselves as the party itself.

Since last winter Britain has been moving through a severe recession, following fairly closely the warning forecasts laid down by Prime Minister Thatcher's government. Unemployment is now higher than at any time since the Depression, and likely to go higher in the bleak winter ahead. Inflation and interest rates are down a little from last spring, but still very high.

This recession is the wringer that Mrs. Thatcher promised when she came into office last year. She's delivering. Her view is that it's the only way to curtail the inflation and revive industrial efficiency. How long will the wringer roll on? To maintain her present complete control over policy, Mrs. Thatcher will probably have to begin showing results sometime this winter. But the fragmentation of the Labor Party, and its rapid leftward drift, give her more time. There is no longer a moderate middle in British political life, offering refuge to any of Mrs. Thatcher's Conservatives who might be tempted to defect. That's good for Mrs. Thatcher's program, and good for her chances of carrying it through despite its mounting costs. But it foreshadows a political system that is profoundly -- and, for Britain, uncharacteristically -- polarized.