How much do you want to bet that in Prodigy, the kids/teens are pressing a bunch of buttons while trying to work out how to fly the thing, and they accidentally send some sort of signal/message to Starfleet? The Admiralty will then, of course, realise that they've not lost their ship and decide to send someone out to retrieve it (that someone being our newly announced Captain Chakotay).
Now, if I had any say in it (I don't, more's the pity), it would be none other than Admiral Janeway who lets our intrepid captain know his mission. She wishes him luck and requests that he makes sure he comes home (yes, at some point, we would see the flash of rings!).
Under the guise of more episodic storytelling, we would mostly follow our rag-tag group of Delta Quadrant natives. We would also get short clips of Chakotay and his team trying to track down these kids who are flying all over the place. Eventually, Chakotay catches up to them, but they freak out and flee. Chakotay has to video Kathryn and tell her that he's seen the ship but is yet to bring it back under Starfleet control and won't be home just yet.
The next time Chakotay catches up with Dal and his "crew", he manages to convince them to hear him out. The kids agree to bring the ship back to Federation space. They get into some adventures along the way, and they (particularly Dal) start to see Chakotay as a sort of father figure. (Of course, the focus is on the kids, so we mostly see things from their perspective).
In my mind, this is over like two seasons. The end of Season One would see Chakotay finally catch up with the wayward ship. And the end of Season Two would have something happen as they make their way back to Federation space that separates Chakotay (and his officers) from the kids. Obviously, Dal and his friends can't leave their new dad behind, so they go after them.
They try to contact Starfleet, and they end up getting hold of Kathryn because they had mimicked what Chakotay would do. They are surprised to see an older looking version of their beloved hologram. Kathryn has already started to fall in love with these kids from her conversations with Chakotay (even though this is the first time she's "met" them), and while she's worried about them and Chakotay, she wishes them luck, nonetheless.
Season Three would see the kids fighting to rescue Chakotay (who we see clips of him and his crew trying to get free of whatever has them) and then being reunited. They would finally get to Earth and meet the Admiral. There would be a reunion between Kathryn and Chakotay, and then Kathryn would extend that to the kids, and they would feel like they have an eclectic but awesome family.
