I am actually in the process of writing a fic for them that kind of shows an abstract development of their relationship during Childe's time in Meropide. From what I could tell at least canonically in the lore, Childe most likely would have spent at least 3 months in Meropide before he disappeared. My internal timeline for them would be like a mix of slow and fast development.

So it would begin with Childe serving, and only hearing of Wriothesley from inmates and the like, tales, rumors, and all of that nonsense. Only to finally meet the Duke himself, who has had some reservations about a Harbinger among his numbers but would amend to treating him just like every other inmate. I believe first impressions would stem from Childe's attempts to spar with Wriothesley, and Wriothesley (obviously) turning him down multiple times. Leaving Wriothesley to just assume that Childe was simply a strange fellow with a pension for trouble. But in moments beyond Childe's harping, they would slowly run into each other more over the early stages of Childe's sentence and be willing to chat about other topics, finding out that conversations between them flow with unexpected ease. Being able to enjoy the friendly banter, coming to strangely enjoy the air of each other's company.

It comes as a bit of a surprise to the both of them to find they'd get along so well, knowing the ways they're used to being treated and perceived by many others. They're both very intuitive people so I'd also assume they'd be capable of very quickly realizing that both of them have very situational behaviors, masks of their personalities, and avoidance when it comes to specific topics or talking about themselves. Realizing how similar they are in that manner just turns into a very natural development of closeness, that gradual increase in thinking about one another and hoping to run into each other. And then wordlessly it would develop into something deeper, without being properly addressed or discussed. Just accepting it for what it is, knowing it's complex and strained yet still wordlessly allowing it to continue because of the strange solace that their relationship would bring.

I think about how they get along frequently... and I'm currently in the process of trying to figure out how their relationship would continue from that point with the aftermath of everything especially now that I have more insight into Childe's whereabouts right after the events and what he could have done during that time. We'll just have to see what I come up with from there but hopefully, you enjoy the ideas I have of them hehe, I'm always happy to share :)