My mother doesn't wait till the end of the story either, before butting in. I get the feeling it's not just this incident that raises the OP's ire, but it's been a building source of irritation for years. At least that is how it is with me.
It'll go, "Part A of this story happened, and...."
"Yes, I know, blah blah blah tangent...."
"Yes, but the reason I was telling about part A is that Part B needs to..."
"Oh, yes, Part B blah blah blah........tangent."
"Wait! That was NOT what I was going to say. So as I was saying, Part B and Part C..."
It's like, constant interruptions, and it's not because I'm long winded. I don't have this problem with other people, and, in fact, I witness her doing this with other people when I really am interested to hear what the other person has to say, but isn't allowed to finish the story.