Joseph: Which kaddish? What day? How old are you?
I’m no rav. So ask your LOR.
This gist that I got from *Halachos of Tefillin*, online, by Shimon D. Eider, seems to say that…
For either: Sounds like you need to keep your Tefillin on if they started kaddish, until after they finish kaddish.
For mourner’s kaddish: Keep the Tefillin on until kaddish is finished, unless for some reason you may need to fluctuate.
…You said “while you’re in middle of putting on Tefilin,” so I don’t know if you meant whether you should continue wrapping/etc, or stop in the middle to wait.
For some reason, I feel like you already know the answer to this.
If so, please do tell 🙂