Okay, so I am forever kind of bummed out that we didn't get a reboot show of Big Time Rush bc I honestly feel like it's so easy to figure out where everyone is starting from?
James is currently living big as a pop star (and was the breakout star of Big Time Rush) but has recently had some kind of scandal or downtown and is struggling to deal with it. But he's the most stereotypically successful of the four and is living pretty in some Hollywood penthouse (and scandal may have involved the tumultuous breakup with an ex-girlfriend that he's currently heartbroken over).
Logan went the Bridgit Mendler route and graduated from some top university with a science degree, and we might honestly steal from the Bridgit Mendler and have him as a CEO of a STEM company. He and Camille aren't together anymore, but she's an incredibly successful and eccentric actress known for her rather campy roles. (Don't worry, they end up getting back together during the show but we gotta have some romantic intrigue and drama here.) She's also still besties with Jo, who is also a very well-known actress but I think she's more known as a television/series actress versus Camille's high profile movies deal.
Carlos is married to spy!Alexa Vega and actually became a spy alongside her, but he just recently was fired from being a spy bc his antics ended up causing just too much trouble. Now his cover, presumably jackass-style daredevil stunts?, is what he's legit doing but is struggling with the transition, which is causing friction in their relationship bc he needs something to genuinely occupy himself.
Katie is a successful PR manager and owns her own firm, despite being young. It's a running gag that people continually underestimate her only for her to have the connection they want/need or already have the client that they were trying to get. She's the one that helps the boys book their first new gig when they come back together as a band (ofc they come back together as a band bc what else is the point of the reboot??)
Gustavo has retired and absolutely hates it and is bored out of his mind but keeps pretending like he's enjoying it until they try to rope him in and then he spouts off about all the joys of retirement and the boys turn to leave and then he immediately fesses up that he hates retirement and don't leave and he ends up becoming partners with Kelly, who is the new head of the studio. Lots of shenanigans come from him having to learn that they're equals now and she's not his assistant anymore. (Also we get some more information on Kelly, including potentially a romance for her and her being Katie's mentor on breaking into the industry when people constantly underestimate you for being a young woman, bc she truly is underrated and deserved more storylines.)
Meanwhile, where's Kendall? Coaching a school hockey time in Minnesota. It is absolutely vital to me that, while the Hollywood/fame perspective is he's washed up and miserable or whatever, Kendall actually enjoys his life, even though it's objectively the most mundane one out of everyone. He was never in it for the fame nor did he really care all that much about it, so he's happy doing what he loves. I'm undecided on if he and Jo are still together or they have a coming back together plot like Logan and Camille, but either way they're still on great terms with one another.
(I have no idea where Lucy is in this tbh, I don't feel like I have as much of a grasp on her character in comparison. My likely answer is that she's a very successful rock star, and her and James had an on-off thing for years that never truly became a relationship but eventually does during the reboot once they're both actual adults. Probably loop her through some music award thing that the boys are trying to work towards, and she's won it previously or is also up for it.)
Kendall, as always, is the connecting piece of BTR bc it's not that the guys aren't still friends with one another but honestly, none of them can really definitively say the last time that they were in a room together was, except for Kendall. The show starts with each of them individually visiting Kendall and being nostalgic about the past, talking about their issues, and kind of not really acknowledging how long it's been since he's seen the other guys. Katie is the last one to visit her big brother and is pretty chill but mostly points out that a) they could make a killing off of a reunion tour b) they've drifted apart and c) they need each other. Kendall is initially hesitant and Katie drops it for the moment, but it's something that plays in the back of everyone's mind for a while. And then, it's been about. what, ten years since BTR's last big tour and there's a "where are they now" article or video (with a mention of how 'will we ever see a reunion between Big Time Rush?') and we see each of them react to it and they all show up at Kendall's house and are throwing out things about how it'd be great and all and Kendall just kinda looks at them until the reasoning eventually becomes truthful with one of them just admitting 'i've really missed you guys'
And BOOM! Big Time Rush reunion setting off onto a new reboot of the show that's aged with fans but is still full of hijinks characteristic of the guys + a little more realistic glimpse at what fame looks like.