No I'm not in a TF2 phase and writing this because I'm writing TF2 fanfiction and genuinely enjoy wholesome child readers...
Ahem, let's start with the RED team.
Scout: Big brother material. He'd put a child up on his shoulders and try to make the kid laugh as much as possible. If the kid is sick or hurt, he's basically the one who sticks beside them to make sure they get better. His nicknames would be "Kid", "Red" (as in from the Boston Red Sox), and "Kiddo". He would definitely not seek help from Spy. If the child is angry or sad, he tries to help out, although he's not super helpful with listening to someone vent. He does still care.
Medic: Essentially the unlicensed pediatrician but he'd also be dealing with the injuries a child would sustain from being with the other mercs. He'd be quite protective whether or not the child is genetically his. If the kid is his, that child would be lab-grown. His nicknames for the child would be "Kind", ("Mädchen", depending on if the kid is a girl or not), and "Child". Whether the child is a boy, girl, neither, or both doesn't matter so much to Medic as long as they're alive and have all their organs. He wouldn't perform needless surgery on the child since they're young, but definitely when they're older, as he has more experience with larger organs and they wouldn't be so delicate. He'd join Pyro for tea parties on occasion. He tries to gauge from his own experiences in his childhood to help the child be comfortable around him, no matter if he plans to experiment on them when they're older.
Demoman: Let's face it, he'd be teaching the kid how to make bombs and pyrotechnics. If the kid were his, he'd forget how he got them but still try to somewhat be a father for them anyway. He would forget the name of the child, so they would have a codename that he'd remember and stick to for some strange reason.
Soldier: He'd mess around with the kid a lot more. Like Scout, the child would be up on his shoulders if they're being taken somewhere. Of course, he wouldn't realize the kid would be there for about a year until someone mentions them one day and he comes to find the child isn't a small mercenary. He would still speak pretty loudly and, depending on how sensitive the kid's ears are, would cause headaches. If the child were his, Zhanna would exist and live in the base with the child. He wouldn't apply himself so much, but he'd appreciate the kid being around.
Spy: He wouldn't touch the kid unless he absolutely needs to, such as fainting, being in a life-or-death situation, being sick, things of the like. He does somewhat care even if the kid isn't his, since he'd be one of the few calm and mostly sane figures in their life. He would definitely teach them how some of his equipment works when the kid is older. If the kid is a girl, he'd call her "Tournesol" and "Lapin" if the kid is a boy. He'd also swap between "Lapin" and "Tournesol" alongside "Milieu" if the child is neither or both.
Pyro: He would very much feed into a child's developing pyromania. He would also not exactly be a parental figure, but more of a sibling/playmate who also happens to gift the kid drawings and, if it's a special occasion, a custom or storebought lighter. The child must be at every small tea party he hosts. BLU Pyro could be a part of those as well. Even if the child grows up, he's still going to see them as whatever age they were introduced to him as, not out of refusing to see them as a growing person (he very much does try to treat them maturely when they're older, although he's not so successful) but because Pyrovision do be like that. He would assign a sign that would mean "Child" or "Kid" in the language he uses with Sniper. He would join Demoman in teaching pyrotechnics and how to handle controlled fire and destruction.
Sniper: He would be the one to teach the kid the language he uses between him and Pyro. He would also join alongside the kid and Medic for Pyro's tea parties. He would also be the one to teach the child how to shoot numerous guns, such as semi-automatic or regular rifles, pistols, and others under the unspoken supervision of Spy. No matter what, the kid is going to be called an ankle biter. If he happens to go on any outings with the child, he would normally be the one to claim being the father unless the child already has another mercenary as a genetic parent. He would genuinely try to help if the kid is going through a tough time, which involves showing multiple hiding places in the base for the child to crawl into and listening to whatever is bothering them, although the advice he'd give might be weirdly worded as he's more of an action taker than an advice giver.
Heavy: He does consider the child family even if the kid is not his. He would call the kid "Ditya" or "Rebenka", which both essentially mean child, although "Ditya" refers to offspring while "Rebenka" refers to the noun child. He wouldn't talk a lot to the child, due to having limited English, but he does enjoy their company. He would happen to spend a lot of time with Medic alongside the child if Medic happens to have them with him.
Engineer: He's not the best with kids, but he'd try. He would try to include the kid in what project he's doing, either by explaining how it works or letting the kid demonstrate if he renders it safe enough. He would also be another mercenary the kid would be available to vent with, as he would try to offer advice that makes a little more sense than Sniper's.
Now the BLUs! Yes, I imagine them both being slightly different although they'd have a mostly similar personality and appearance.
Scout: He'd be pretty careful with the kid, much more careful than RED. His nicknames would be the same as RED Scout's, although he would not use "Red", he would use "Blue" (as in from the Toronto Blue Jays, he's still a fan of the Red Sox though). Mostly he would be serious in some situations, although he does try to make the kid laugh whenever he can.
Medic: He not only treats the kid's injuries, but he also scolds whoever they got the injuries from. Unless it's Heavy, he would just tell him not to do it again next time. He would not participate in BLU Pyro's tea parties as much as RED Medic does, although everything else is the same as RED.
Demoman: He'd still try his best and teach the kid how to make bombs, he'd just be a little drunker. He wouldn't apply himself so much, but he'd still be around.
Soldier: He'd actually try to be a little quieter for the kid's sake. He'd apply himself as much as RED Demoman would. How the hell would he be genetically related to the kid, though? BLU Zhanna, I guess. Or he thought they were a raccoon and essentially picked them off the street. Wouldn't that mean he... nope, nevermind. No. He would not be related to the kid. Just illegal adoption. No freaky shit happening here, no sirree.
Spy: The exact same as RED Spy, he'd just be pretty late to being introduced since he's a head in RED Medic's fridge. Also, he'd be way more emotionally available. He did raise a kid before, after all.
Pyro: An absolute sweetheart. He'd try to invite RED Pyro to his tea parties, and he'd try to participate in RED's as well. He's the only class who would be friends with someone on the other team who is the same class as him. He would also be on the exact same wavelength in terms of behavior with RED Pyro, other than the fact he'd expose the child a lot more to fire without Spy being around to discourage him.
Sniper: He'd be a little less patient with the child, but he'd remind himself that they're trying their best and they're still learning. He wouldn't be as available as RED Sniper is in terms of venting, but he offers a helping hand whenever he can.
Heavy: Heavy but BLU and it takes a little longer for him to get used to a child being in the same place as him. He'd still think of the kid when the base is under attack and try to take them to safety alongside BLU Scout.
Engineer: He'd try to teach the kid some recipes to make up for being private about his projects until he thinks the kid won't get hurt from unpredictability. He does not want Medic up his ass again. He'd also have a beard for some reason, comic stuff.
Update: Rewording and grammar. I might add stuff in the future btw
Update 2: So BLU Soldier has a oneshot dedicated to him accidentally adopting a child via thinking it's a garbage panda. It's not on this app/site, but I just wanted to let you know.

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My Headcanons
FanficI'd be on my deathbed and I would still fight for these headcanons.