Aborting would have dashed flamboyant billionaire Branson's hopes of beating rival Jeff Bezos, whose own flight to space was scheduled a few days later.
The spaceship floated down on three giant parachutes before firing a retro thruster, sending up a cloud of sand as it gently landed at one or two kilometres an hour.
If successful, Jeff Bezos' Blue Origin programme plans two more passenger flights to space by this year's end.
Scientists are worried about the potential for longer term harm as the industry is poised for major growth, particularly impacts to the ozone layer in the still poorly understood upper atmosphere.
Blue Origin announced that instead of a USD 28 million auction winner launching with founder Jeff Bezos on Tuesday, Dutch runner-up Oliver Daemen will be on board.
It will be the first launch with passengers for Blue Origin, which like Richard Branson's Virgin Galactic plans to start flying paying customers in the months ahead.
The spaceflight should take place at 8:30 am Mountain Time (1430 GMT), after overnight weather conditions forced a 90-minute delay. Virgin Galactic is livestreaming the launch on its website.
The winged rocket ship will soar from New Mexico — the first carrying a full crew of company employees. It will be only the fourth trip to space for Virgin Galactic.
The trip is 60 years overdue for Wally Funk, who was one of the Mercury 13 -- the first women trained to fly to space from 1960-1961, but excluded because of their gender.
Saturday's successful bidder beat out some 20 rivals in an auction launched on May 19 and wrapped up with a 10-minute, livecast frenzy.
Bezos said in early February that he was stepping down as CEO of Amazon to spend more time, among other things, focused on Blue Origin.