The U.S. invasion of Grenada has created serious political problems for Prime Minister Margaret Thatcher's government, undermining her claim to a special relationship with the Reagan administration and even raising questions about the imminent deployment in Britain of American cruise missiles.

Thatcher's inability to dissuade Reagan from mounting the operation and the government's clumsy handling of the crucial question of when it learned that an invasion was in prospect has damaged the authority and credibility of her leadership. The Conservatives, a senior Thatcher aide said today, "can be nothing but unhappy" about the situation.

In a detailed statement to an emergency debate in Parliament, Foreign Secretary Geoffrey Howe sought unsuccessfully to contain the trouble. His account of the chronology of Britain's involvement showed not only that the government knew more and earlier about the prospects of an American-led intervention than it had previously conceded, but also that its communications with the United States and Caribbean nations involved were apparently confused and incomplete.

Howe, who again refused to condemn the invasion despite Britain's misgivings, complained only that the "extent of the consultation" by the Reagan administration was "regrettably less than we would have wished."

The indignation here over the Grenada invasion seems to be a combination of wounded pride because the Reagan administration ignored Britain's advice about a Commonwealth country that used to be its colony and a strong feeling, even among Conservatives that the action taken by the United States violates international law.

"Can Reagan be trusted?" headlined one newspaper tonight, capturing the widespread feeling of politicians and the press that the United States had breached its relations with Thatcher by its approach to the crisis. If the prime minister was unable to stop Reagan from moving on Grenada, opposition spokesmen declared throughout the day, she would be less able to prevent him from firing U.S. nuclear missiles based on British territory if he believed it in American interests to do so.

To now accept a U.S. pledge to respect Britain's right to veto use of the missiles is to live "in a dangerous fool's paradise," said Enoch Powell, a respected veteran member of Parliament who represents the Northern Ireland Unionists. "Anyone in office who entertains that delusion is in no position to serve the security of this country."

"There is no credible analogy" between the missiles issue and the U.S. rejection of British advice on Grenada, Howe insisted in Parliament.

But privately, government sources acknowledged that the emotional uproar over Grenada has revived powerful instincts for Britain to obtain some physical control over the 160 missiles scheduled to begin arriving next month--the so-called "dual key"--that will be difficult to contain.

Howe, who has borne the brunt of defending the government's case, was severely criticized by Tory members of Parliament at a closed meeting last night for his ineffectual performance in the face of opposition attacks, according to reliable reports. His speech today was intended to refute allegations that he had been deliberately misleading in previous parliamentary statements about the affair, one of the most serious charges that can be made against a British politician.

On Monday, Howe said he was aware of no U.S. "intention" to intervene on the island and yesterday he added that word of the invasion plan was received in London several hours after that initial statement.

Today, however, Howe said that the British ambassador in Barbados had learned on Friday that "some Caribbean heads of government" were pressing other countries in the region to ask for military help to restore order in Grenada. "We promptly took steps and instructed our embassy in Washington to ascertain how the United States might respond to such an approach," he said.

He said that on Saturday the government was advised that the Organization of Eastern Caribbean States had "decided to put together a multinational force and to call on friendly governments" for assistance. He said that the United States reported receiving a firm request to join and that it had concluded to proceed "very cautiously."

Sunday, Britain was told it would be asked to join such an operation.

"On that day, we were assured by the United States government that we would be consulted immediately if the United States decided to take any action," Howe said, and it was on the basis of those assurances that he told the House of Commons he knew of no U.S. intentions to invade. Only that evening was Britain formally requested to join in the operation, precipitating a series of Anglo-U.S. exchanges that culminated after midnight in Thatcher's telephone call to Reagan urging him not to act.

Although repeating his reservations about the invasion, Howe refused to say how Britain would vote on the issue at the United Nations. To jeers from Labor and other opposition, he said, "What has happened must not be allowed to weaken the essential fabric of our relationship with the United States."