Is it permissible for someone whose time is constrained between Fajr prayer and leaving for work and study to pray in congregation with his family at home and then they depart, knowing that he is keen to pray in the mosque with the congregation on other days?

Preserving congregational prayer in the mosque is a religious requirement. However, if going to the mosque delays the questioner from his work or causes his family to miss school at the appointed time, then this is an excuse that permits him to be absent from congregational prayer in the mosque. He will still receive its reward if he prays with his family in his home.