The idea of using the leaves of a Merkle tree to store alternative Bitcoin scripts dates back to Russell O’Connor in 2012. The idea of using Schnorr signatures instead of ECDSA has been discussed for at least as long. So you could say that what we will collectively call BIP-Taproot (BIP 340, BIP 341 and BIP 342) has been in the making for over eight years at this point.
However, there have been different iterations of how to apply these ideas and what we now consider BIP-Taproot was first proposed on the Bitcoin dev mailing list by Greg Maxwell in January 2018.
The draft of the Schnorr BIP (what became BIP 340) was first sent to the Bitcoin dev mailing list in July 2018.
The draft of the Taproot BIP and the Tapscript BIP (what became BIP 341 and BIP 342) were sent to the Bitcoin dev mailing list in May 2019. The BIPs were all allocated BIP numbers in January 2020.
Jonas Nick laid out the full timeline of BIP-Taproot progress here.
At the time of writing (December 2020) the Taproot PR has been merged into the Bitcoin Core repository but there has been no Bitcoin Core release with activation parameters set and hence Taproot is not currently activated on the Bitcoin blockchain.