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-Bukhaari (378) and Muslim (417), from
the hadeeth of Anas ibn Maalik and Abu Hurayrah (may Allaah be pleased with
them), according to which the Messenger of Allaah (peace and blessings
of Allaah 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 Allaahu liman hamidah, then
say Allaahumma 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 Allaah 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 Allaah be pleased with him) who said: When the
Messenger of Allaah (S) said “Sami’a Allaahu liman hamidah,” no one
of us would bend his back until the Messenger of Allaah (S) had gone down in
prostration, then we would go down in prostration after him. Narrated by
al-Bukhaari (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 Allaah knows
best.