In the sentence 最後の言葉の時だけほんの少し俯いたけど when she says she got depressed or disappointed, it refers to how many words of the last sentence she said (or maybe the whole sentence as she had said 3 long sentences in a row).
Generally, when someone is emotionally affected by 最後の言葉, it means not the exact the last word, but the last whole sentence or some of last sentences which convey one meaningful content.
We use the similar expression like this as
In this case general translation will be like;
The bride cried a little only at the last part of her father's speech at the wedding ceremony.