Bungie revealed some of their future plans with Destiny 2, which included the release date or window of the Transmog system, depending on how you look at it.
Destiny 2 will get the Transmog system in Season 14, according to the latest blog post from the developers. Technically, this is the release date but since we don't know the exact date of Season 14's release, we decided to just call it a release window as we know when that will happen, more or less.
Season of the Hunt, also known as Season 12, is the one in progress at the time of writing and it's expected to wrap up on February 9, 2020. This is the same date Season 13 will commence and is likely to last three or four months, meaning Season 14 will probably start in May or June 2021. In other words, it is one of these two months that we can expect the Transmog system to pop up in.
Bungie also offered some of the early screenshots of the system , one of which you can see on the image above but noted that this is work in progress and subject to change.
One thing we are certain many will notice is that the images don't show any sort of UI improvements that would help us manage the shaders.
Still, it is worth keeping in mind that Bungie might introduce something like that as they do eventually release features community requests. One of the best examples is crossplay, which is coming in 2021 as well.