Everything that happened in canon still happened. I've never read the cursed child so this goes divergent from canon after the epilogue in book 7.
Albus is indeed in slytherin and hes been isolated by his brother and everyone around him. years past and even lily ignores him when she gets to hogwarts. he doesnt have friends and hes fighting severe depression.
one day he stumbles upon a dusty room that looks like it hasn't been open in decades. Theres a covered portrait in the room and, curious, Al opens the curtains. Before him is the glaring image of Lord Voldemort, looking angry and evil and all that jazz. Al flees.
But he keeps thinking about it. Cant stop thinking about it He has no one to talk to about it. He's drifted away from his mother and father and he knows his mother was disappointed in him when he was sorted into slytherin. he decides to go back to the room.
voldemort is still there, but he only glares at him, not speaking. So Al talks. he tells voldemort about who he is, his family life, how isolated he feels. he can't tell anyone else and this is a portrait that quite obviously has no access to anywhere but this room.
But then voldemort starts talking back.
they fight at first, al not wanting to get along with the monster that tried to murder his father. The monster that endorsed committing genocide. But as his loneliness starts to slip away, he starts to listen to the things that voldemort is saying. He knows that hes being manipulated. He's a lonely teenager. it doesn't take much for him to be eating up everything voldemort is saying.
voldemort starts teaching him spells, giving him advice on homework. the end of term comes and al is suddenly faced with the fact that he will have to leave voldemort behind. and he can't do that. so he pries the portrait from the wall, shrinks it, and smuggles it out of hogwarts.
he keeps the portrait on him at all times. a constant companion. voldemort convinces al to help him get a body back. (The portrait is a horcrux created at the final battle at hogwarts. no one but him knew it existed) voldemort gets his body back, but he doesnt look like himself. because of the way the ritual is performed, he takes on Al's age (let's say they're going to be 6th years when school comes around again) he looks faintly like himself, same hair color, but his face looks slightly different and his eyes are blue instead of the brown they used to be. he looks distinct enough from his older self that he could pass as his own grandson.
before al agrees to do the ritual, he made voldemort swear on his magic that he won't continue where he left off. he can't kill people, muggle or otherwise, unless they have come after him with the intent to kill him. voldemort agrees to stay by albus' side and to try and change his views on the condition that he gets a body.
al starts to pretend to receive letters from a friend. his father and mother are shocked he has someone who wants to write to him, but then they are ecstatic. al explains that they did a pen pal program in one of his classes (and they did--his original pen pal never wrote him back) and that the boy is a Beauxbatons student.(you can decide tom's name but Theiry Richelieu is french-ish and has meaninf behind both first and last name; up to you) Harry and Ginny encourage him to invite his friend over and al agrees. they meet voldemort and are none the wiser. harry actually becomes quite fond of the young frenchmen. too fond.
al watches, half in horror, as his father and his 'pen pal' become close. he can see how they play off of each other and al sees it coming before they do. during that summer, voldemort and harry develop feelings for each other. voldemorts actually starting to shift his views the longer he stays in al's presence and this shift is for the good. but now his dad is one step away from cheating on his mom with the man who tried to kill him. the man that al helped bring back.
al now needs to decide what he's going to do. should he let this flirtation turn into something more? he doesn't want his mother getting hurt, but their relationship is almost completely fallen apart due to her standoffish nature thanks to his sorting. voldemort has mentored him and helped him by being his friends for his entire fifth year. he's conflicted. (At this point voldemort genuinely cares about the boy; it weirds him out. add on to the fact that he's seeing harry potter in an entirely new light and he's positively shook)
and thats all i've thought up. i imagine al and voldemort are exposed at some point, maybe by a jealous and spiteful ginny?