Is the prayer of the congregant valid if he initially intended to complete one rak'ah with the imam who had omitted a prostration in his previous prayer, then changed his intention to complete the two rak'ahs of Fajr after the imam began praying two full rak'ahs?

If the Imam forgets a prostration (sujud) in the last rak'ah of the Fajr prayer and remembers after the taslim, then if the interval is not long, he should say Allahu Akbar, prostrate, then recite the tashahhud and perform the taslim. Afterwards, he should perform the prostrations of forgetfulness (sujud al-sahw) and then perform the taslim again, without performing an additional rak'ah. However, if the interval is long, or he spoke for a reason unrelated to the prayer, or he invalidated his wudu', he must restart the prayer from the beginning. The determination of whether the interval is long or short is based on common understanding and custom, not a specific duration.

As for the ma'mum (follower) who entered with the intention of completing the prayer, then changed his intention to starting a new prayer without a takbirat al-ihram, his prayer is not valid. Transitioning from one intention to another during prayer without a takbirat al-ihram invalidates the first prayer and does not validate the second. What he should have done was to intend the Fajr prayer, say the takbirat al-ihram, and then pray two rak'ahs, or remain on his initial intention and separate from the Imam after the Imam stands for the second rak'ah and then perform the taslim, or wait for him and perform the taslim with him.