There are multiple consequences. Firstly, his relationship with his father will be forever changed. He'll no longer be able to trust him in the same way. Secondly, his relationship with Ladybug might be affected. She may be more cautious around him, not knowing if he'll side with his father. Thirdly, Adrien will have to face the public's reaction if the truth ever gets out. He could become a target or be shunned. All of these aspects make for a very intense and emotional storyline in fanfiction.
He might be shocked at first. In many fanfictions, Adrien would feel a mix of disbelief, anger, and sadness. He has always known his father as a strict but loving parent, and finding out he's Hawk Moth, a supervillain, would completely shatter his view of him.
It could be Gabriel Agreste. He has been a suspect in the original series for various reasons. He is often absent and has a lot of secrets. So in the fanfiction, it would make sense if he was finally revealed as Hawk Moth.
Common themes are transformation, not just in the sense of the Miraculous transformations but also in the characters' personalities. Hawk Moth might transform from a pure - evil figure to something more complex. Another is the exploration of the Miraculous powers. Fanfics often dig deeper into how the powers work and how they can be used in different ways against Hawk Moth. And secrecy is a big theme. Keeping identities secret, Hawk Moth's secrets, and how those secrets impact the story.
Perhaps Adrien becomes Hawkmoth because he's been pushed to the edge by Ladybug's rejections. As Chat Noir, he's constantly in love with her but feels unappreciated. The pain of this unrequited love could twist his perception of good and evil. He might think that if he were in control as Hawkmoth, he could make Ladybug see him in a different light. He could use the power to create situations where Ladybug has to rely on him more, or at least understand his pain.
It could be Ladybug. If Adrien turns villain, Ladybug would be the natural antagonist as she represents the good side. Their relationship would be completely flipped from what it was before. She would have to try to stop him from causing more harm, all while probably still having some feelings for the old Adrien.
One great fanfiction is 'The Hawk Moth's Redemption'. It explores an alternate universe where Hawk Moth starts to question his motives. Another interesting one is 'Ladybug and Hawk Moth's Unlikely Alliance' which has a unique plot of the two having to team up for a common cause. And 'Hawk Moth's Hidden Past' delves into his backstory and how he became the villain we know.
One possible plot could be that Ladybug and Chat Noir finally track down Hawk Moth's lair. They discover that Hawk Moth is someone they know in their daily lives, like a teacher or a neighbor. This revelation shocks them, but also gives them an edge in the fight as they now understand his motives better.