For the speaker, time passes and nothing lasts forever, and the speaker is all too aware of the passage of time.
For the swans, however, "their hearts have not grown old." They are "unwearied" and, it seems, unchanged. They are mysterious and beautiful, unaware of time.
Another word for battered is often scarred, and it seems as though they have been through hardships from their dialogue and feelings, so I think the likely answer would be scarred.