How can the prayer be corrected after the Imam forgot the first Tashahhud and the congregants remained seated, thinking that he was sitting, then some of them followed him into bowing while others completed their prayer individually?

Congregants who fall behind their Imam due to a valid excuse should perform what they missed and then continue with him for the remainder of the prayer. Their prayer is valid. If the Imam precedes the congregant by one or two pillars due to an excuse such as inattentiveness or not hearing, the congregant should perform what they missed and then continue with their Imam. As for someone who continued with the Imam without making up the missed pillars, they must perform an additional rak'ah after the Imam's salām; otherwise, they are required to repeat the entire prayer. And whoever intended to separate from the Imam and completed their prayer individually, their prayer is valid.