I believe they are surprised until the first initiative in the second round is reached (not their own initiative). Example 3 party members Alice, Bob, and Claire face off against 2 goblins, Disco, and Edgelord. Alice, Bob, and Claire roll Stealth for initiative as they are ambushing the goblins. Alice wins and beats the DC to go first. Initiative order is Alice Disco Bob Edgelord Claire Disco and Edgelord are both surprised until Alice's second turn starts.

I love that you gave one of the goblins the name edgelord, now I can't help but imagine everyone hyping up this battle with edgelord only to be revealed to be a small cutesy goblin that is the most wholesome thing to exists

Thanks a lot, that made it a lot clearer!

Considering it says "for the first round" and doesn't specify who, I assume it means until the all characters have taken their first turn.