The present tense of 「お花しおれてっちゃった」 is 「お花しおれてっちゃう」.
Let me put 「ちゃった」 or 「ちゃう」 part aside for now, please. So I can give you a clearer explanation of why the heroine chose the word 「～ていく」(「～てっ」 form in this sentence) instead of 「～てきた」 here.
By the way, this「〜てっ」 means 「〜ていって」 as you translated. And I think that this 「〜ていって」 is 「〜ていく」＋「て」. So, my explanation uses 「〜ていく」 below.
The heroine used 「～ていく」 because she was seeing the future at that time. She was feeling the worst future was coming or some of the worst future had already come and more of them were coming. What she wanted to express was that things were getting worse and worse. So, 「～ていく」 matches her feelings more than 「～てきた」 which mainly expresses what happened in the past.
Then a question comes up: why did she use 「ちゃった」 instead of 「ちゃう」? The answer is that she used the past tense in order to emphasize "It's too late". I think that the nuance of 「お花しおれてっちゃった」 in this case is "it's toooo late! There is no longer any way to save these lives."
In general, 「お花しおれてっちゃった」 sounds strange a little in this kind of situation. 「お花しおれてっちゃう」, 「お花しおれちゃった」, 「お花しおれてきちゃった」(as explained in other user's answers) etc. would be more appropriate as a common Japanese sentence. But she said 「お花しおれてっちゃった」. To find out why, let's talk about the real world (not textbook world).
Sometimes people, especially kids, just create a new word or a new way to combine words in order to express and emphasize their real feelings. Existing words or ways are not enough for them.
Kids often use this kind of technic to exaggerate what they really want to tell, because they want to impress people especially their parents by using limited vocabulary (poorer vocabulary than grown people). And mothers usually understand this kind of kids' efforts and see through the exaggeration.
After the heroine uttered the line, her mother said 「あなたずーっと握りしめてるんだもの。おうちについたら水切りすれば大丈夫よ」 in the movie. Her mother was obviously thinking that the heroine was exaggerating it too much. The nuance of her mother's line is like "No surprise. There is a cause then there is its effect. And actually, it's NOT too late."
I would say that the heroine's line and tone were kind of very childish. It's great because we, viewers, can see the difference between childish her at that time and grown her after having experienced the special things in the movie.
So, I think that the heroine's reference point was "now", but she used the past tense 「ちゃった」 in order to emphasize "it's too late." It's not a common way to say. It's the heroine's way to emphasize her real feelings as a child to her parents.