When should the one who is praying behind the imam bow? When the imam says takbeer, or straight after the takbeer, or when?

What is prescribed for the one who is praying behind the imam is to follow his imam, not to do anything before him or at the same time as him, or to delay doing it. Rather he should do things straight after his imam does them. 

This is indicated by the report narrated by al-Bukhari (378) and Muslim (417), from the hadeeth of Anas ibn Maalik and Abu Hurayrah (may Allah be pleased with them), according to which the Messenger of Allah (peace and blessings of Allah be upon him) said: “The imam is only appointed to be followed, so when he says takbeer, then say takbeer; when he bows, then bow (and do not bow until he bows); when he says Sami’a Allahu liman hamidah, then say Allahumma rabbana laka’l-hamd. If he prays standing, then pray standing.” The phrase in brackets was added in the version narrated by Abu Dawood (603). 

The words of the Prophet (peace and blessings of Allah be upon him): “when he bows, then bow and do not bow until he bows”, and the same with regard to prostration, indicate that the person who is praying behind the imam should not start to move to the next part of the prayer until after the imam has done it. So he should not bow until the imam bows, and he should not prostrate until the imam prostrates. This is stated clearly in the hadeeth of al-Bara’ ibn ‘Aazib (may Allah be pleased with him) who said: When the Messenger of Allah (S) said “Sami’a Allahu liman hamidah,” no one of us would bend his back until the Messenger of Allah (S) had gone down in prostration, then we would go down in prostration after him. Narrated by al-Bukhari (811)and Muslim (474). 

Al-Nawawi said in Sharh Muslim: This hadeeth shows that this is one of the etiquettes of prayer and it is Sunnah for the one who is praying behind the imam not to start going down (for prostration) until the imam has placed his forehead on the ground. End quote. 

Based on this, what is prescribed for the one who is praying behind the imam is to follow his imam straight after he has moved to the next part of the prayer. So what counts is the action of the imam, not his takbeer. This applies to those who can see the imam. As for the members of the congregation who cannot see him, then they should follow his words, and they should start to move to the next part of the prayer after the imam finishes saying the takbeer. See also the answer to question no. 33790. 

And Allah knows best.