The Thatcher government today approved an extensive cost-cutting restructuring of Britain's military forces during the 1980s that will reduce Army, Navy and civilian defense manpower and reshape the Royal Navy and its role in NATO.

The savings will enable Britain to afford needed high technology weapons systems, modernization of its independent nuclear deterrent with Trident submarine-launched missiles developed by the United States as well as deployment of British naval task forces alongside U.S. ships outside the NATO area in places like the Indian Ocean.

The biggest changes will be made in the Royal Navy and the way it would help the U.S. Navy defend Atlantic supply routes against Soviet submarines in time of war. Britain now intends to rely less on big, expensive aircraft carriers, destroyers and frigates to protect troop convoys across the Atlantic in wartime and more on cheaper, smaller ships, aircraft and hunger-killer submarines.

The number of British surface ships assigned to NATO duty will be cut from 59 to 50, with eight of the remaining 50 on "standby" status in port with skeleton crews. All big ships, including two aircraft carriers now under construction will be finished, but at least one existing carrier and a number of destroyers and frigates will be taken out of service before they require expensive refitting.

The biggest savings will come from the closing of one big naval dockyard and the shrinking of another. Manpower will be cut in the next four years by up to 20,000 in the civilian work force of the Defense Ministry, 10,000 in the Navy and 7,000 in the Army.

Although 2,000 soldiers will be brought home from the British Army on the Rhine in West Germany by reorganizing its command structure, Britain will still have 55,000 troops on the NATO front line there. British officials considered much bigger reductions, informed sources said today, but they were dissuaded by NATO treaty commitments and arguments that NATO "would start to fall apart" if large numbers of British troops were pulled off the front line.

The axe will fall instead on the Royal Navy's big surface ships in the Atlantic, despite anticipated displeasure of British and American naval commanders. British officials acknowledged today that the American naval presence in the Atlantic probably will have to be increased because of the British changes. They said details of how the two countries will share naval defense roles in the Atlantic in the future still have to be worked out.

British Defense Secretary John Nott, who supervised the defense reorganization approved today by Prime Minister Margaret Thatcher's Cabinet, explained it to U.S. Defense Secretary Caspar Weinberger in Washington last weekend. Nott said today that Weinberger questioned him closely about the cutbacks in surface ships, but was pleased that Britain has still committed itself to overall defense spending increases of 3 percent above inflation over the next four years.

I think they accepted that we've done as best we could," Nott said of his recent consultations with American, West German and NATO officials.

Nott pointed out that Britain's level of defense spending, measured either per capita or as a proportion of gross national product, is exceeded in NATO only by the United States. But because of the escalating cost of military technology and the many defense roles Britain tries to perform in NATO, he said, the savings and reorganization were necessary.

"We have a defense program which is unbalanced and overextended," Nott said. "We cannot go on like this."

Britain had been forced to cut back on deployment of ships outside the NATO defense area during the past two years, threatening a long tradition of global sailing by what once was the world's largest navy. Nott said his changes in defense spending will enable Britain once again to send "a substantial naval task group" on extended assignments in the South Atlantic, Caribbean, Indian Ocean or farther east, often in coordination with the United States and other allies.

He said Britain also would continue preparations for a small rapid deployment force of two specially trained Army battalions that could assist the planned U.S. Rapid Deployment Force outside the NATO area.

To critics who have questioned the $10 billion price tag of Trident, which will replace Britain's present Polaris submarine-launched nuclear missile system, Nott said, "No enhancement of our conventional forces could possibly prove of equal deterrent value" as the warheads that can be fired from a single Trident submarine, which could strike 128 different targets in the Soviet Union.

British officials said the hard decisions on defense priorities and need for savings and greater efficiency forced on Britain by the cost of weapons such as Trident will have to be faced by the other NATO nations, including the United States. They said the Reagan administration's big increases in defense spending would eventually be eaten up by the same increases in military technology costs.