There could be some of Harry's friends who might play a role. They could be the ones who encourage Harry to meet the Queen or who give him advice on how to behave in front of her.
Well, it could be that the Queen, being a symbol of order in the Muggle world, has to deal with the sudden appearance of Harry Potter and the chaos that magic can bring. Maybe she initially mistakes his magic for some sort of advanced technology and there are a series of misunderstandings that lead to an interesting adventure. Harry could introduce her to the concept of Hogwarts and she might be fascinated by the idea of a school for magic.
The main characters would be Harry Potter and the Queen. Harry, as the hero from the wizarding world, brings his magic, courage, and sense of justice. The Queen represents the Muggle royalty and all the power and traditions that come with it. There could also be some secondary characters from the wizarding world like Hermione and Ron, who might give advice to Harry on how to interact with the Muggle Queen.
Well, he might use his magic to show the Queen some amazing things. Maybe he'd conjure up a beautiful display of Hogwarts' magic, like floating candles or flying books. And the Queen would be both surprised and fascinated.
Probably when Harry shows the Queen his magic wand for the first time. Just the idea of a real magic wand in the hands of a wizard in front of the Queen would be thrilling.
Well, perhaps Harry Potter could use his magic to solve some sort of mystery in the Queen's palace. Maybe he'd help protect a valuable magical artifact that was in the Queen's possession without her knowledge of its true nature.
Well, it could be that Harry Potter, while on a magical mission, somehow gets transported to Star City where Oliver Queen operates. Oliver, being the vigilant Arrow, at first mistakes Harry for a threat. But then they realize they both fight for justice in their own ways. Harry could use his magic to assist Oliver in taking down some really tough villains that the Arrow alone couldn't handle, like some dark magic - infused criminals.
Well, it could be about Sonic having an encounter with Queen Aleena. Maybe they team up to face a common enemy in the fanfiction. It might also involve Sonic learning something from Queen Aleena or vice versa.
It would probably be Harry Potter himself. He's from a world full of magic and mystery, and seeing him interact with the very real and regal world of the Queen would be fascinating.