Assalamu alaykum. Suppose I being Imam say takbir and the followers behind me also say takbir and start zuhr salah. But after a while I remembered that i had made a mistake in my niyyah. So I say takbir again but in low voice and start praying from the begining. But the followers dont know that fact. Is the salah of mine and the followers valid?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

Your prayer is valid if you had intended to interrupt the first prayer, and you will get the reward of an individual prayer and not a congregational prayer unless more people came after that and prayed behind you, and you intended to be their Imam, then the reward of a congregational prayer will be written for you in this case.

With regard to the prayer of the worshippers, the basic principle is that the prayer of the people led in prayer is invalid if they said the Takbeeratul-Ihraam before the Imam. This is what you mentioned in the question. But the scholars said that the prayer of the worshippers led in prayer is valid here (in the case which you mentioned in the question) because this is something that is hidden from them; so their prayer is not invalidated due to them saying the Takbeeratul-Ihraam before you.

Tuhaftul-Muhtaaj, which is one of the Shaafi'i School of jurisprudence books, reads:

“Al-Hannati and others said, if he (the person leading the prayer –Imam) said the Takbeeratul-Ihraam, and then he says the Takbeeratul-Ihraam again silently, in a way that the people led in prayer did not hear him, then this does not harm the validity of praying behind him, even if the prayer of the Imam is void, because this is something hidden and there is no indication for knowing it.” [End of quote]

The footnote of Ash-Sharwaani ‘ala at-Tuhfah reads:

“His saying: ‘Even if the prayer of the Imam is void’, then the invalidation applies to the second if there is no nullifier to the first one, such as his intention of interrupting it…...” [End of quote]
The footnote of Ash-Sharwani ‘ala at-Tuhfah also reads:

“As for the Imam, if he did not intend to interrupt the first one, for example, between the two Takbeers, then his prayer is void because of its termination by the second Takbeer. Otherwise, his prayer is valid individually because the people led by him did not renew their following of him; so if some other worshippers come after him and prayed behind him, and he intended to lead them in prayer, then he gets the reward of a congregational prayer……” [End of quote]

Nihaayatu az-Zayn, which is also one of the Shaafi'i School books, reads:

“If the person led in prayer says the Takbeeratul-Ihraam after the Imam and then the Imam says again the Takbeeratul-Ihraam silently with intention, because he doubted about the first Takbeeratul-Ihraam, then this does not harm the validity of the prayer of the people led by him, because this is something hidden (to them) and there is no indication for knowing it.”

Allah knows best.