I'll look later, but I believe Belthesar says something along the lines of "you are really dead Serge", and the context of it implied that he put more weight on the history of Another World than that of Home World.
If that is so, then Kid can't appear in 1010 AD.
But I still don't follow this logic. I don't see why she couldn't appear in 1010 AD in the Another World timeline. Let's look at it like this, with Another World A and Another World B designating a change to the timeline after Kid's time travel.
Another World A - Kid time travels in 1020 AD.
Another World B - Kid arrives in 1010 AD, dimensions split.
I don't see why she couldn't appear in the past of
both dimensions, since we know from the story that it apparently wasn't her direct action that caused Serge to live. She set up the 50/50 chance of his survival somehow. At the exact time of the dimensional split, everything was duplicated, including Kid, and the only difference between the two dimensions was that Serge died in one and lived in the other.
Furthermore, I'm 100% certain that it doesn't say anywhere in the script that Kid only appeared in Home World, and that would be a rather strange situation - wouldn't it? If she only appeared in Home World then that would mean that the dimensions had to split
before her arrival back in time. And that doesn't make sense, unless I am overlooking something huge.