President Reagan, responding to criticism that his budget cuts were endangering the lives of the nation's coal miners, reversed himself yesterday and said he would restore some budget and personnel cuts for the Labor Department's Mine Safety and Health Administration.

The president said he would release $2 million in withheld 1982 mine safety funds in order to lift a hiring freeze that reduced the number of mine inspectors by 9 percent, from 1,389 to 1,264, between the end of fiscal 1979 and fiscal 1981. Reagan also said he would amend his 1983 budget to include an additional $15 million for mine safety.

The administration had planned to ask Congress to keep MSHA funding for fiscal 1983 at $141 million, the same amount budgeted for 1982. But the president's revised request would raise proposed spending for mine safety to $156 million for 1983, White House officials said yesterday.

Twenty miners have died in work-related accidents since Jan. 1, compared to nine for the same period last year. A total of 153 miners were killed in 1981, the highest annual toll since 1975.

United Mine Workers President Sam Church Jr. had charged that the deaths were related to reduced inspections caused by budget cuts.