Is it better to pray the congregational prayer at the beginning of its time, or to continue studying or consulting, then pray the congregational prayer, with evidence?

If there is a mosque near you, you must pray there in congregation, as this is a religious requirement. The larger the congregation, the greater the reward.

If there is no mosque near you, then you should pause your studies or consultation and pray at the beginning of the prayer time, as it is recommended to hasten to prayer immediately after its time begins. This is supported by the hadith: "Prayer at its early time," and the hadith of Abu Dharr: "Pray the prayer at its [appointed] time. If you then catch it with them, pray, for it will be a supererogatory prayer for you." According to Al-Nawawi in his commentary on Sahih Muslim, this hadith encourages praying at the beginning of the time, and it is recommended for a follower to pray alone and then pray with the Imam if the Imam delays it beyond its time.

It should be noted that delaying prayer beyond its time is not permissible except for an excuse such as forgetfulness or sleep.