Joseph, who is from Northamptonshire, vowed to continue to enable Linus's "insane ideas" - as he dressed as Princes Diana and Gwyneth Paltrow for their other shows Diana: The Untold and Untrue Story, and Gwyneth Goes Skiing.
Linus, who's from Sweden, described Joseph as the "sweetest most ridiculous person" and said he couldn't wait to create "more stupid fringe shows" together.
Both sets of parents joined their sons on stage where Linus's mother Elisabeth Ljunggren said they were happy to share their day.
"Love is a gift to cherish and care for, every day in good times and bad, in sickness and health," she said.
"We love you both to the moon and back."
Fellow performers including Sooz Kemper, Emily Lamey and theatre company Recent Cutbacks provided the entertainment, and guests were invited to donate to charity or buy tickets to their shows in lieu of gifts.
The happy couple had just over an hour with their guests in the outdoor bar before they had to leave for that day's performance.
Meanwhile with US and UK dates in September and December, there's little time for a honeymoon either.
"I'm not sure there'll be time for a real honeymoon for a little while at least but performing at the fringe is very much a honeymoon in itself, very relaxing," says Linus.
"Maybe we'll climb Arthur's seat. That seems like a good way to celebrate our Scottish wedding."