Good question. "Long winded" is often about something someone said (It's been an hour—that was a long-winded explanation!) or about the person themselves (She's a long-winded speaker, her lectures go on for hours!). So here in your example it simply means: a long way of saying "next week" i.e. instead of saying "next week," the person has said something much longer instead of just "next week." This must have happened before your example, because I don't see anything which would make sense with "a long winded way of saying next week." Perhaps before the conversation we're seeing, person A said something like this: "We'll move there as soon as I can get the car sold, which I think is going to happen tomorrow because I've got a buyer who is very interested. But we also need to wait for Tim to finish the school year, which won't be til next Tuesday. Oh, and Max wants to see Clara one more time, maybe that'll be Wednesday, not this Wednesday but next..." = next week. Except that was a really long-winded way of saying it.

Long-winded usually means a story that is too long. It generates a lot of wind (breath), I guess to get it all out. Anyway, the context seems to be that before this quote even started, A was telling a long story, the point of which was to say that his family was moving here next week. (It might have been something like "what an ordeal getting air tickets, the flights were too expensive and no direct flights were available and then we couldn't arrange the truck pick-up" and so on. Person A in the last line says in effect a short version of something like this: "Sorry, [the big long unnecessary story I just told you was a] long-winded way of saying "next week". Does that make sense?