NBC News anchor Tom Brokaw, conspicuously absent from the nation's television screens since before the beginning of the current Mideast hostage crisis, has cut short his African vacation and was en route late yesterday to Beirut, hoping to become the first network anchor on the scene.

"If all goes well," an NBC spokesman says, Brokaw will be heard from Beirut by telephone hookup on this morning's edition of the "Today" show and will be seen reporting on tonight's edition of "NBC Nightly News."

Sources at the network were reluctant to be specific on Brokaw's travel plans for fear for his safety, but they confirmed that Brokaw, whose vacation was not to have ended until July 9, was scheduled to land in Beirut at 2 a.m. today (Washington time) and begin work soon after. Brokaw had been all but inaccessible as he safaried through the African bush with his wife and two daughters, spokesmen said, but Monday he phoned NBC News President Lawrence K. Grossman from Nairobi and volunteered for the mission to Beirut.

He told Grossman, who Friday cut short his own vacation in Greece to return to New York, "I want to go to Beirut. I want to report this story," according to NBC sources. Until now, Brokaw had been the only network anchor not on the air during the hostage crisis that followed the hijacking of a TWA jet. But NBC sources said they neither urged Brokaw to return to New York nor coaxed him into going to Beirut. It was Brokaw's idea to go, but it was "very much a joint decision" that he do so, the source said.

Terrorists in Beirut who hold the hostages have repeatedly fired gunshots in the direction of reporters and photographers at the scene. Yesterday, Dan Rather reported on CBS that all media were asked to leave the airport and that more shots were fired over their heads before they did. The situation is volatile and unpredictable. NBC News officials expressed some nervousness over Brokaw's decision to go to Lebanon himself.

Brokaw reportedly called Grossman yesterday to say, "Guess where I am?" He was in Cyprus, about to get on a charter flight to Beirut. Brokaw had been on a flight from Nairobi to London and was going to double back to the Mideast on another flight, but he "talked his way off the airplane" when it stopped for refueling in Cyprus, an NBC News executive said, and was thereby able to shorten travel time.

The Brokaw redeployment came on the very day that ratings for last week's network newscasts were released. They revealed that while both the CBS and ABC evening newscasts, with their regular anchors at the helm, gained substantial numbers of viewers over the previous week, viewership of "NBC Nightly News" was up only slightly. "CBS Evening News With Dan Rather," the usual front-runner, led the three networks with a 12.3 rating and a 26 percent share of the viewing audience (the highest Rather rating since December 1983), an 11 percent increase over the previous week, before the crisis began.

"ABC World News Tonight" was up 7 percent over the previous week, with a 10.2 rating and 21 share. But "NBC Nightly News," with Roger Mudd substituting for Brokaw, was up only 1 percent, with a 9.9 rating and 21 share.

David Poltrack, head of research at CBS, estimated that the number of households watching the "CBS Evening News" increased by 1 million (from 9.4 million the previous week to 10.4 million last week) and that the number of homes tuned to all three networks rose from 25.8 million to 27.5 million. Normally in the summer months, news viewing levels drop rather than rise. During the same week of 1984, the total viewership for the three network news programs was 23.3 million homes, Poltrack said.

The three-network share went up from 64 percent the previous week to 68 percent last week. CBS claimed that of the increased viewers, its "Evening News" got 60 percent, ABC got 35 percent and NBC only 5 percent. But an NBC spokesman scoffed at the figures, saying they were based on trifling data.

He also said the decision for Brokaw to go to Beirut was made before the ratings were known.

"We believe our coverage was complete and competitive using our tremendous bench strength, and we still very much believe that," the NBC source said. But he conceded that the presence of an anchor in Beirut would add a particular distinction to the network's coverage.

When Brokaw spoke with Grossman on Monday, he told Grossman that even while on safari, "I spend half my time looking at animals and the other half tuned to the BBC," listening for reports on the continuing crisis. Brokaw reportedly told Grossman, "It's an important story and a major crisis, and I think I should report it." Starting today, he will.