Henley, who finished all 18 holes in the first round, and Oosthuizen ended the first day atop the leaderboard at 4-under 67. Oosthuizen was among 36 players who failed to finish due to the 90-minute fog delay at the start, which caused the field to run behind schedule.
Oosthuizen, who had two holes remaining in his first round, and the others will finish the opening round Friday morning and then head into their second rounds.
Francesco Molinari and Rafa Cabrera Bello completed their first rounds and were tied for third at 3-under par.
Two-time U.S. Open winner Brooks Koepka, meanwhile, shot 2-under 69 on the first day for a share of fifth. It was his sixth consecutive round in the 60s at the U.S. Open, with 11 of his last 14 rounds being in the 60s at the tournament.
"I've just got a good game plan, focused, I know what I'm doing, and I don't try to do anything I can't," Koepka said. "It's just all about discipline in a U.S. Open. That's the gist of it."
Xander Schauffele, Hideki Matsuyama and Jon Rahm also were among the six-man group at 2-under 69.
Defending U.S. Open champion Bryson DeChambeau had to play the last five holes in 2-under to salvage a 73. He carded four birdies and six bogeys.
Round 1 will continue simultaneously when the second round begins Friday at about 9:45 a.m. EDT.