We miss the prayer while we are in private lessons, what should we do?

Delaying the prayer beyond its prescribed time due to a private lesson is forbidden and is considered a major sin, for Allah Almighty says: (Indeed, prayer has been decreed upon the believers at specified times) and His saying: (Then, there succeeded them a progeny who neglected prayers and pursued desires; so they are going to meet Ghayy), and repentance is obligatory for whoever does that.

As for missing the congregational prayer, the preponderant opinion is that it is individually obligatory for able men. Among the proofs for this is Allah Almighty's saying: (And when you are among them and lead them in prayer, let a group of them stand with you), and the Prophet's (peace be upon him) saying: "I was about to order firewood to be gathered, then order the prayer to be announced, then order a man to lead the people in prayer, then go to men who do not attend the prayer and burn their houses down upon them." Therefore, you must repent and change the lesson's schedule.