All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
The worshipper does not have to be silent when he hears the Imaam say "Allaahu Akbar". There is no harm in the worshipper completing the verse or phrase he is saying while the Imaam is saying "Allaahu Akbar". What one is commanded to do is to move right after the Imaam moves without delay. Saying a word or two and the like is not considered delaying and does not invalidate the prayer.
As for an obligatory Thikr, it is obligatory to perform it in its entirety at the time in which one is commanded to recite it. For instance, it is obligatory to say "Subhaana rabbiya al-‘atheem" once in Rukoo‘, so one has to say it in full while bowing. One also has to recite Al-Faatihah while standing. It is impermissible to recite any part of it after standing. The same applies to the first Tashahhud; according to the scholars who hold that it is obligatory, it has to be recited in full while one is sitting; otherwise, one would be leaving out an obligatory act of the prayer.
Allaah Knows best.