[When Is Sending Blessings upon the Prophet (May Allah bless him and his family and grant them peace) Obligatory]

The answer — and Allah is the Granter of success and assistance — is that sending blessings upon the Prophet (May Allah bless him and his family and grant them peace) is obligatory, as indicated by the verse of Al-Ahzab, and there is no disagreement regarding the obligation of sending blessings upon him (May Allah bless him and his family and grant them peace) in general. The Sunnah has been transmitted regarding the obligation of sending blessings upon the Prophet (May Allah bless him and his family and grant them peace) during the prayer, and it is also obligatory in the Friday sermon; and this is an explanation of what came in the verse of Al-Ahzab.
As for the hadith: “Whoever I am mentioned in his presence and he does not send blessings upon you …” and so on, no one — as far as it appears — has acted upon its apparent meaning; thus its meaning is referred back to the obligation of sending blessings upon the Prophet (May Allah bless him and his family and grant them peace) during the prayer. Accordingly, the one who performs the five daily prayers has indeed sent blessings upon the Prophet (May Allah bless him and his family and grant them peace) whenever he is mentioned (May Allah bless him and his family and grant them peace). And Allah knows best. And all praise is for Allah, Lord of the worlds, and may Allah bless Muhammad and his pure family.
As for outside the prayers and outside the Friday sermon, sending blessings upon the Prophet (May Allah bless him and his family and grant them peace) and his family is recommended in all cases, whether he is mentioned (May Allah bless him and his family and grant them peace) or not, due to what has come in the Sunnah encouraging and urging it.

Source: Min Thimār al-ʿIlm wa al-Ḥikmah vol.3