Q. An ovel during shiva that doesn’t have a minyan at home and comes to Shul, when he davens for the amud do they say Tachanun? Should someone else take over and he goes out so the others will say Tachanun?

A. The reason we omit tachanun in the ovel’s home is not to increase the forces of judgment already present on the ovel. (Levush). Mishna Berura (131: 20) rules that if the mourner prays in shul, the attendants don’t have to follow him and they recite tachanun, but he does not (Oruch Hashulchan ibid. 15, see Sdei Chemed vav: 4). The reason is that only when a choson is present, people identify with him and would like to join him in his simcha (Shiurey Knesses Hagdolah 15) Some Poskim maintain that the congregants omit tachanun also (see Kaf Hachaim 131: 60 and Yalkut Yosef – Nefilas Apaim 13)

If he leads the services (see prior question) most Poskim agree that no tachanun is recited (Gesher Hachaim 1: 20: 3, Halichos Shlomo – Tefilah 11: 10, Piskey Teshuvos 131: 16)
Horav Shlomo Miller’s Shlit”a opinion is that the ovel should let someone else take over the omud after finishing the repetition of the amida, so the congregants can recite tachanun, the ovel omits saying tachanun, and returns to the omud for kadish.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a