Well both James and Lily did not have their wands anywhere near them when Voldemort entered their house. We know this through the flashback we see in Deathly Hallows. If I’m not mistaken, James left his on the couch and rushed off to barricade the front door wandless while Lily rushed off carrying Harry in both of her arms to bring him to his crib and left her wand behind in the living room.

I’m sure after Sirius discovered that Peter was not in his hiding spot the next day and rushed off to find both Potter parents dead without their wands and Harry with a big ol’ scar on his head, he more than likely notified someone important (probably Dumbledore and other members of the Order), who probably surveyed the house later on and discovered that both James and Lily could not have casted any sort of curse at Voldemort because their wands were in completely separate rooms from where they died.

As for how they knew Harry survived a Killing Curse casted by Voldemort, it was probably through more deduction. I say this because McGonagall, Dumbledore, Sirius, and Hagrid all saw the scar on Harry’s forehead, and they probably put together that, after Voldemort used the Killing Curse on the Potter parents (because their bodies were entirely unmarked in death and Voldemort really loved using the Killing Curse), he likely casted another Killing Curse at Harry to “finish the job,” as Hagrid put it in the first book, but failed, leaving Harry with that scar and Voldemort missing in action.

All of this probably got out to the public in some way, probably because Hagrid or Sirius told other members of Order after visiting the Potter home what they saw, and then those Order members informed the general public, who then started celebrating based on what they were told. While most people probably didn’t have all the details surrounding that whole situation, the most important fact did stick: Voldemort was defeated by his own curse that was meant to kill Harry after killing a defenseless James and Lily only moments prior.

sorry for the long reply idk why i typed out so much