I think I figured out how to really start on here. Gonna give you guys some Linked Universe headcanons, because I think that would be a good start.
Time: It is a famous headcanon for him, but I see him being one of the most chaotic Link next to Wild, he grow up with Fae beings after all. He might look like the most responsible Link, but don't let it fool you. Time will play pranks and little trick, that he learned while living with the Kokiri and faries, on everyone without them realizing that it was him, especially if someone pissed him off.
Time also doesn't really know what he is doing, despite seemingly leading the whole chain. He tries to hide how clueless he really is with what he is doing and that he seeks help from Warriors at times to make sure he does everything right.
Twilight: I headcanon that his shadow sometimes looks like a wolf, even if in his Hylian form almost like it tries to show what he is/can turn into. At times his shadow also seems to move weird, like something or someone is inside it and Twilight gets reminded of Midna when that happens.
Twilight can't play any instrument, so every time the chain plays some music he turns into Wolfie and howls to be part of it.
Wild: He absolutly loves to cross dress and constantly does if their with the Gerudo's no matter in what timeline the chain is in. Sure he was embarrassed the first time the chain saw him in his Gerudo outfit, but that was only because they teased him when seeing it.
The chain has one rule: Don't make Wild angry! This rule was made after they teased him too much and he put Goron spice in the food, after that they learned it is better to not make him angry if you want to eat something good and not spicy.
Legend: We know he is a little grumpy guy and is a bunny in the dark world, so I can see him stumping his feet like an angry bunny if he gets annoyed too much or just really angry, he doesn't even have to be in his bunny form for it and in general has bunny instinc even in his Hylian form. The chain thinks it is kinda cute whenever they see him stump like an angry bunny and Sky and Twilight have to keep themselves from laughing out loud because they know why Legend does it.
Everytime they are in Legend's time, he gets pulled into a big hug by Ravio and gets forced to sit down and take a break by him. Both always end up cuddling together with Legend telling Ravio of everything that happened and giving him some stuff he found to look after it.
Hyrule: It is comman to say that Hyrule is part fariy and can turn into one, so I headcanon that he constantly stays near Time when there is danger because something on Time pulls his fariy sense to him and wanting to make sure that Time is alright. Hyrule is very confused on why it happens and trys his best to hide it, but Legend quickly found out and now both of them are talking about it at times.
Hyrule constantly picks up flowers and makes pretty flower crowns for everyone and even managed to pick the favorit flower from each Link for the crowns, he even made one for Marlon, Ravio and Aryll.
Wind: Wind despite being the youngest Link in the chain, is the Link that constantly tries to play the big brother for everyone because he is not used to get treated like the younger sibling and hates it. Wind just doesn't like being the youngest and treated like a child because of it, he is a hero after all and doesn't need to be treated like that!
Wind loves to swim. Everytime the chain is somewhere near water Wind is the first to go in and stay in it for hours swimming. It reminds him of home, even if it is not fully an island.
That's it for now, I don't have any headcanons for the other boys right now, but I still hope you like some of the headcanons