The Shulhan Aruch (Orah Haim 38:5) writes that a mourner does not wear Tefillin on the first day of mourning. If, for example, one’s family member passed away, Heaven forbid, on Sunday night, and the funeral takes place on Monday, the mourner does not wear Tefillin on Monday, and he resumes wearing Tefillin on Tuesday.
Often, the burial takes place the day after the passing. Hacham David Yosef (contemporary), in his Halacha Berura (vol. 3, p. 215), brings different views among the Poskim as to when the mourner resumes wearing Tefillin in such a case. The Maharit Sahalon (1559-1638) maintained that if the burial takes place the day after the death, then the mourner wears Tefillin after the burial, that same day. Since this is not the day of the loved one’s passing, Tefillin is required. Others, however, including the Elya Rabba (Rav Eliyahu Shapira, Prague, 1660-1712), Mas’at Binyamin (Rav Binyamin Aharon Selnik, Poland, 1530-1620) and Peri Megadim (Rav Yosef Ben Meir Teomim, 1727-1792), disagreed. In their view, the day of the burial is treated like the day of death, and so the mourners do not wear Tefillin that day. As for the practical Halacha, Hacham Ovadia Yosef ruled that given the difference of opinion among the Poskim, in such a case one should wear Tefillin on the second day, but in private, and without reciting the Beracha.
An exception to this rule is when the family member passed away on Shabbat, and is buried on Sunday. Since the burial cannot take place on Shabbat, Shabbat is not considered the first day of mourning. Therefore, Tefillin are not worn the following day, Sunday, even after the burial. This is the ruling of Hacham Ovadia Yosef (Yabia Omer, vol. 2, Y.D. 27:8), based on the Kanhag ("Kenesset Ha’gedola," Rav Haim Benveniste, Turkey, 1603-1673).
On the second day of mourning, the mourner must not put on his Tefillin until after Netz Ha’hama (sunrise). This is because fundamentally, Tefillin are not worn even on the second day, but due to the rule of "Miskat Ha’yom Ke’kulo" ("part of the day is like the entirety of the day"), the mourner waits a few moments into the second day, and then resumes wearing Tefillin. Thus, on the day after the burial, the mourner must ensure not to put on Tefillin before sunrise. In the wintertime, sunrise can be as late as 7:20am or so, and so mourners must be cognizant of this Halacha and make a point of not putting on their Tefillin on the day following the burial until after sunrise.
Summary: Mourners do not wear Tefillin on the first day of mourning; they resume wearing Tefillin after sunrise on the day following the burial. If the burial did not take place on the day of the family member’s death, then the mourner puts on Tefillin after the burial, in private and without a Beracha, unless the family member died on Shabbat and was buried on Sunday, in which case the mourner does not put on Tefillin until after sunrise on Monday morning.