Idk, just trying things out...

Henry: it depends on the timeline. He's the character that changes the most, either becoming worse, better, the same, or anything in between.

So-

Ceo Henry: Leafon. Henry was incredibly close to getting friends, but betrayed them. I don't think he is able to evolve based on friendship, because of that, and Leafon is the eeveelution with the highest defense (defending himself from possible betrayal which causes him to lose the opportunity to make friends). Plus, there's that Animal Crossing fail in there which makes me think of leaf and plants.

TT Henry: Slyveon. Slyveon is a friendship evolution and I think is the hardest eeveelution to get. It's also not an espeion (day) and not an umberon (night). Somewhere inbetween. I think that would fit a Henry that is friends with Charles and Ellie well.

VH: Umberon. Friendship and Henry 100% evolved when slaming his paws against the window of the escape pod. Since he was surrounded by darkness and mourning his best friend, he evolved into an umberon. A permanent way to remember Charles.

BRO: Espeion. Friendship with no Ellie. Charles will naturally get him to evolve towards an espeion, due to friendship and not being a dark loving thief

JB: Jolton. Mostly because he'd have to a stone evolution in that path because... you know. Also Jolton has the most speed so, he's quick to steal a overlarge gem and run away while people chase after him.

PP: Umbreon. Sticks with Ellie and keeps his ruby. Plus, he's definitely going to steal again. (remember the dark type is actually called the "evil" type in Japan)

TR: Espeion. Henry evolves when Dmitri knocks out Ellie, and because he wants to protect his friend, he evolves on the spot and uses a psychic attack to throw Dmitri off the rocket.

SSR: Glaceion. Because space is cold and he kind of relaxes a bit in the ending. Jolton would work as well as the whole route and the previous is nothing but quick time events.

FM: Vapeorn. I was thinking Jolton, but Henry isn't on the offensive this time. He's mostly on the defensive. He's only attacking because the clan kidnapped him. Vapeorn has the highest hp, so more protection and not in a "Henry's guarded" way but more in "Henry is lucky" way. Also, it would be so screwed up as part of Right's revenge to Henry is force him to take the water stone, as Henry permanently altered Right's body so it's only fair for him to do the same thing to him. (jokes on RHM cause Henry used his new tail to help him steer his body in space) (might make a good fic)

CG: Umberon. He evolved when the heist is over at the sunset. Theives.

LNE: Leafon. Mostly because of pressumed dead, but this is the only timeline I don't think Henry was ever discovered to not be dead. I think he laid low long enough in either the jungle to evolve camoflauge. That or Glacion for the same reason but he just stayed in Canada long enough for that to happen. I choose Leafon for the defense.

MBH: Flareon. Henry is on friendly terms with Charles, but I wouldn't call them best friends. I think Henry evolved into a Flareon because of it's high attack and how he goes for bounty's is by attacking, so why not?

TCW: I think he stays as an eevee, or at least for now. Nothing of this timeline really shouts that he would evolve. In fact it's the only ending where the game doesn't know if it's a mission complete. It ends on a cliff hanger. Maybe become a flareon for the attack or a Vapeorn if Ellie beats him to flareon. (more on that later)(also if Henry beats her to it, she'd evolve into a Vapeorn lol)(Charles looking at a shamrock: "Guess I got to be a Leafon now")

TK: Espeion. Evolves during the raid, after Reginald pulls him up. Reginald's approval even though it scared Henry so bad, causes him to evolve.

T4L: Glacion for the special attack as Henry's loyalty towards the clan. Plus the presumed dead thing. I believe loyalty to a whole cause doesn't count for friendship points. Plus he was holding on to that helicopter for probably hours. In that span, he probably evolved with "righteous" anger in his heart against the gov't.

R: Henry should be a flareon. The fire bursting in his heart at the betrayal and the attack stat, but Dr. Vin is a good doctor. Evolving in Henry's state would hurt him. She takes away the fire stone and tells him "he's still recovering and she'd need to make new cybernetics for his evolution, which could kill him in that time". Reginald not only tried to kill him, but cost him his ability to evolve. Something all eevees instinctively want. It's also tragic at the ending part, where Henry- having so many options to evolve, to be who he wants to be, dies as an eevee. Unchanged.

Aren't there other characters? Oh yeah!

Ellie: Espeion*. I think she only evolves into that during Convict Allies when she hits Dmitri with a stop sign. Yeah, it's night, but Espeion fits her better because the force is definitely a psychic type thing. The other timelines, I don't think she does as she's stuck at the wall being depressed and friendless or Henry's betrayal causes her to regress a bit and be stuck (though I do think in most GI timelines she either becomes an espeion due to joining the toppat clan or possibly befriending Charles but that's my headcanon). TCW can be tragic with her evolving into a flareon or vaperon where she looses herself to her rage and does irreparable damage to herself. (do you think eeveelutions get evolution dysphoria?)

Charles: Umberon. Not for any good reason though. I think it's more of a sign of Henry's influence and friendship with Charles. Making him less of a yes man in the military. Rebelling a bit. Plus it make TT cool where the team is all friendship evolutions. Also, Charles definitely evolves into an umberon in VH right before he dies. There's also a bit of irony in it, where the happy go lucky pilot evolves in a dark type, but maybe that's the point. Where the side of Charles he tries to hide comes out, when he accepts it as a part of who he is. Doesn't make him evil, it's just accepting who he is. (Also in timelines where he doesn't befriend Henry, he either stays as an eevee or Rupert's possibly friendship with him is enough to evolve him into an umberon).

Reginald: Probably Slyveon to be honest. Reginald seems like a very pampered individual, his friendship with Right is so strong he could evolve via friendship, his gayness, and fairy is a strong type plus special defense (for his paranoia). It just fits him.

Right: Flareon is probably the best as he might be a bit hot tempered and acts with emotions. He could've purposely evolved into a flareon for the strength (he's a very good attacker in canon) so he can protect Reginald and is a good way to hide his past with evolution. Showing that he's a different person now, as he's no longer an eevee. I'd say it be before canon, but he'd probably hate Henry even more if was still looking for a fire stone when he got chopped in half, and is no longer able to evolve. (Makes the vaperon thing more in character, plus vaperon could've been what he was going for previously before he let that life behind so double punishment for Henry)

Dave: I'm not sure, but I'm going towards Umberon. He has a relationship with the night, I also think he would evolve after the game, due to being put in the brig and it being traumatizing enough that he couldn't evolve. But once he's out and rescued either by Rupert, Charles, or Henry, he'd start to slowly recover and become an umberon through their efforts.

Rupert: Flareon. After leaving the police force, he takes a fire stone as he wants to get stronger and do more good. Kind of another metaphor for going into a new life with evolution. He chooses Flareon because it's high attack and his own temper. (same as Right).

Galeforce: Leafon, because he just has that green leafy vibe man. Also, the defense could be translated to stubbornness where no one can convince him that throwing Henry at the Toppat Clan might not be a good idea. Kind of insane for that, so I think he needs a lot defense to be able to do crazy stuff like that. Jolton is also a good pick for his insanity.

Sven: Vaperon. Kind of because fanon associates him with sharks, but also high hp, where he becomes the leader after the clan suffers a huge blow as the most important division in the clan is arrested. Like the clan is on low hp, and needs someone with high hp to tank the blow. He's also swimming in paperwork lol.

Burt: He's aiming for Slyveon but it's taking forever. Why slyveon? Well, I was thinking jolton thanks to his job, but he's not that energetic. I think judging by the comics on his wall and how his shoes are mismatched, he would want to be something nobody would expect him to be at first glance. A slyveon. He's working hard for it in his off time, but he doesn't have a solid friendship to help him with it. It won't stop him though.