I have a new Linked Universe headcannon about Tears of the Kingdom until Jojo come out with an official one.
Unbeknownst to the other links, Wild has been hearing the voice of his Zelda and keeping it a secret. Eventually, a portal opens up in the middle of the night, Wild, mesmerized by the voice of Zelda, walks through, returning to his Hyrule, where he lives out the events up until the end of Tears of the Kingdom. He has fully upgraded his arm and reclaimed the Master Sword.
In the Linked Universe, the other links wake up, but without the usual smell of breakfast ( As Wild wakes up at 5 am ) the realize that their compatriot is missing. Even Twilight with both his kenn sense of smell an connection to Wild, can't tell where he has gone.
After 6 days and 6 nights, they decide to pack up and move on, hoping that Wild simply left and would follow eventually. In a last ditch effort, Wind tries to use the pirates charm to contact Wild. Suddenly light blue streaks appear in the form of a running figure with an oddly familiar sword. Eventually the blue disputes, leaving a collapsed Wild with longer hair, an armored tunic, a strange arm, and a master sword with a blade that Sky barely recognizes. ( I feel that because Totk's Master Sword is reformed and Wild at this point has proven himself to easily weild it, it can be used)
Wild awakens on a bed surrounded by the other links, with Sky holding his reformed sword. With a concerned look on his face
I should probably clarify a few things, Wild has regained all of his memories from botw and all of zeldas from totk. The sage abilities are not able to be used.