The Tur (Orah Haim 149) explains that there are two customs with regards to returning the Sefer Torah to the Hechal on a weekday. One custom, as followed by many Ashkenzim, is that once the reading is completed, Kaddish is recited by the Hazan, the Sefer Torah is returned to the Hechal, and then “Uva Letzion” is recited. The Arizal supports this custom because he posits that once the reading of the Torah is completed, the spiritual lights that emanate from it are extinguished and the Torah should therefore be returned immediately. The second opinion given by the Tur, which was the custom in Spain, is to return the Torah after the reciting “Uva Letzion”. This latter custom is further subdivided into two practices, with some Sepharadim reciting Kaddish Titkabal after the Torah is returned to the Hechal, and others, based on the Avudraham and the Kolbo, reciting Kaddish Titkabal while the Torah is still out.
Furthermore, the Shulhan Aruch (Orah Haim 25:13) writes that the custom is to avoid removing one’s Tefilin while the Torah is still out. The Mishna Berura (ibid:57) says that this is alluded to in the verse (Micha 2:13) “Vaya’avor Malkam Lifnehem V’Hashem BeRosham” (lit. “and their king passed before them, and Hashem was at their head”), with the Tefilin Shel Rosh being symbolic of Hashem. The ideal state of a Jewish man is to be crowned with the Tefilin, and therefore if one must remove one’s Tefilin while the Torah is still out, it is proper to do so while the scroll is covered up so as not to reveal one’s bare head. Nevertheless, the Ben Ish Hai (Od Yosef Hai, Parashat Haye Sara, § 2) says that this only applies to those scrolls that are covered in a velvet cover, commonly used by Ashkenazim and Moroccans. One, however would be permitted to remove one’s Tefillin beside a Torah that is placed in a hard case since the scroll is permanently covered.
Summary: One should not remove one’s Tefilin while the Torah still out. If one must do so, one should wait until the scroll is covered.