Interruptions When Putting on Tefillin
Question
What do I do if I’m putting in my tefillin and they’re saying Kaddish?
Answer
Shalom!
Thank you for your question.
Virtually all interruptions are forbidden when putting on tefillin. One reason for this is that according to many authorities the wearing of the “shel-yad” (arm tefillin) and the wearing of the “shel-rosh” (head tefillin) constitutes a single mitzva. As such, any interruption would be an interruption in a single, on-going mitzva.
The ban on interrupting when putting on tefillin includes even mere gesturing, winking, or other ”sign language,” and of course, it goes without saying that any speaking is forbidden, as well. Even remaining silent for no good reason might be an interruption, as well! We see form here that one should put the tefillin on promptly, one after the other with no interruption whatsoever.
Getting back to our question, yes, even answering kaddish and kedusha is forbidden while putting on tefillin. If one hears such congregational prayers as one is putting on tefillin, one should remain silent and just listen to the responses.
One putting on Rabbeinu Tam tefillin, which is an added stringency and/or custom of many people though not truly required, is permitted to respond to kaddish and kedusha. But if one did so, one should re-adjust the arm tefillin and then re-adjust the head tefillin in order to symbolize putting on the arm tefillin and the head tefillin one after another without interruption.
Source
OC 25 and commentaries