The general practice is to not wear tefillin in the morning on Tisha BAv (OC 555:1). However, after chatzos many of the mourning practices which are only customs are lifted. As such, people wear tefillin at Mincha (see Mishnah Berurah ad. loc.).
However, I have seen many congregations that delay Mincha until the end of the fast. As a result, they delay putting on tefillin until then. My impression is they don't put them on earlier at home. But, why not? There's a concept of זריזים מקדימים למצוות, the alacritous perform Mitzvos at the earliest opportunity (see for example Pesachim 4a). Why is the general practice to wait until even a late Mincha? I could see the idea that it's better to pray with tefillin on, but why not put them on earlier? Or enforce Mincha Gedolah?