Well, it depends on how the author of the fanfiction writes it. Maybe Sirius would be initially angry. He has a certain view of Harry as a young wizard he cares about, like a surrogate family member. So, seeing Harry in such an intimate situation might make him feel like he has failed in some way to guide Harry properly. On the other hand, it could also lead to a more understanding Sirius, who realizes that Harry is growing up and tries to support him in a more nuanced way, like helping him understand the boundaries of relationships.
The relationship could become strained. Sirius might start to view Harry differently, with a bit of wariness. He might not trust Harry as much as before in terms of making good decisions. But it could also lead to a more open relationship where they talk about things they previously avoided, like relationships and feelings.
There may be such fanfiction in some niche corners of the internet, but this goes against the original nature of the Harry Potter series and is also a rather strange and inappropriate concept. J.K. Rowling's original works don't have any such plotlines, and most official and mainstream interpretations stay true to the original characters' relationships and story arcs.
After being caught, they could be forced to keep their relationship more low - key for a while. This might involve sneaking around more carefully. They may also have to deal with the opinions of others, like Harry having to face the Weasley family's concerns about their relationship. Ginny would have to assure her family that Harry is the right person for her.
Yes, there might be such fanfiction out there in the vast world of fan - created works. However, this kind of story often takes a rather unorthodox and perhaps controversial approach to the characters. Fanfiction can cover a wide range of themes and interpretations, but this one is likely to be quite different from the typical Harry Potter stories we know.
Maybe Sirius gets help from some powerful wizards or witches who believe in his innocence. They might testify on his behalf in front of the wizarding authorities. Sirius also could show his strong emotional bond with Harry. He has always been there for Harry in spirit, and now he can be there physically to raise him, which convinces the authorities to give him custody.
In some fanfictions, it could be that they are caught by Hermione and Ron, which leads to a lot of teasing and awkward moments. For example, Ron might make a big fuss about it while Hermione tries to be more diplomatic.
In many fanfictions, if Harry and Ginny were caught by Sirius, Sirius might be initially protective and concerned. He could question their actions, especially if they were in a situation that seemed dangerous or inappropriate for their age. For example, if they were out after curfew or in a restricted area. Sirius might give them a stern lecture about being responsible and the importance of following rules, while also understanding their adventurous spirits as he was quite the rebel himself when he was young.
One major problem is that it goes against the nature of the relationship between Sirius and Harry in the original 'Harry Potter' series. Their relationship is more of a mentor - mentee or a father - son like bond, not a romantic or sexual one.