President Reagan, as required by the the Gramm-Rudman-Hollings budget law, yesterday activated the mechanism that will slice $23 billion in federal spending unless he and Congress can agree on a tax increase or budget cuts by Nov. 20.

The cuts will hit programs ranging from weapons to education grants.

Office of Management and Budget Director James C. Miller III said the cuts would require furloughing FBI agents, letting go as many as 1,500 State Department employes, cutting back the National Science Foundation, making major layoffs at NASA and delaying hiring air traffic controllers.

At 8 a.m. yesterday Miller told reporters that "the president will veto a tax increase."

But by late afternoon, after meeting with economic advisers and watching the gyrating stock market, Reagan read a conciliatory statement to reporters.

"I am willing to look at whatever proposal they {Congress} might have," he said when asked whether he would reconsider a tax increase. But he said a few minutes later that he still opposes raising taxes.

Congress passed the automatic spending-cut provision last month to force the president to choose between a tax increase or a drastic reduction in some of his most cherished defense programs.

Once Reagan reluctantly signed the law, the spending reductions became inevitable unless he and Congress agree on a way to reduce the fiscal 1988 budget deficit by $23 billion.

Until yesterday afternoon, it appeared likely the president would choose to sacrifice the military rather than accept a tax increase.

If the cuts take place, most domestic programs would be slashed by 8.5 percent, Miller said yesterday. The exceptions are interest payments, Social Security, Aid to Families with Dependent Children and a few other programs designed to aid the old or needy that were exempted by Congress.

All military programs would be cut by 10.5 percent, except for those covering manpower, which were exempted by the president.

Under the law, the cuts would be evenly split between domestic and military programs -- $11.5 billion from domestic programs and $11.5 billion from the military. Because Reagan spared military manpower accounts, the percentage cut would be higher in the remaining portion of the military budget.

The cuts would go account by account, from a $5,000 reduction in the $62,000 mileage allowance for the vice president and senators to a $58,000 reduction in the $683,000 budget of the United States Institute of Peace.

The House Democratic Study Group said that "while some high-priority programs are exempt, many are not. Items that would be subject to the full 8.5 percent reduction include AIDS-research and -prevention programs, other biomedical research law enforcement, air traffic control, compensatory education, student financial aid, Head Start, Older Americans Act programs, low-income housing and various agricultural programs.

"The cut also would be applied to all legislative branch operations, including members' allowances and committee staffs," the study group noted.

The Children's Defense Fund said the automatic cuts would be "fairly massive." About $100 million could be cut from proposed Head Start spending, which could translate into programs affecting about 40,000 children.

Compensatory education programs to help disadvantaged children learn reading and math could lose more than $350 million, which could mean serving 600,000 fewer children, according to the fund.

Personnel directors in several major agencies said they have not yet estimated possible layoffs.

When the House Government Operations Committee conducted a survey two years ago -- before the Supreme Court struck down an earlier version of the law's automatic spending-cut mechanism -- the Commerce Department envisioned furloughing 15,367 employes for 11 days and the Agriculture Department thought it might furlough 23,400 people for 21 to 24 days.

Most other agencies answering the questionnaire predicted some furloughs or reductions in force. At that time, a 7.3 percent spending cut was envisioned.