Speaking about when the Direct could happen, a brief (maybe) analysis that touches on two points
1) the day of the week
Funnily enough, there's some consistency here when looking at major presentations and I have to thank
MondoMega for noticing it; I don't understand the reason behind it, but there's a pattern that can be traced back by looking at the Directs of the Switch era
2017
Fire Emblem Direct January 18th, 2017 - Wednesday
Ninendo Direct April 12th, 2017 - Wednesday
Arms / Splatoon 2 Direct May 17th, 2017 - Wednesday
Pokémon Direct June 6th, 2017 - Tuesday
Nintendo Spotlight: E3 2017 - Tuesday
Splatoon 2 Direct July 6th, 2017 - Thursday
Nintendo Direct September 13th, 2017 - Wednesday
Animal Crossing Pocket Camp Mobile Direct October 24th, 2017 - Tuesday
Xenoblade Chronicles 2 Direct November 7th, 2017 - Tuesday
2018
Nintendo Direct Mini January 11th, 2018 - Thursday
Nintendo Direct March 8th, 2018 - Thursday
Nintendo Direct E3 2018 - Tuesday
Super Smash Bros. Ultimate Direct August 8th, 2018 - Wednesday
Dragalia Lost Mobile Direct August 29th, 2018 - Wednesday
Nintendo Direct September 6th / 13th, 2018 - Thursday
Super Smash Bros. Ultimate Direct November 1st, 2018 - Thursday
2019
Nintendo Direct February 13th, 2019 - Wednesday
Pokémon Direct February 27th, 2019 - Wednesday
Super Mario Maker 2 Direct May 15th, 2019 - Wednesday
Pokémon Direct June 5th, 2019 - Wednesday
Nintendo Direct E3 2019 - Tuesday
Nintendo Direct September 4th, 2019 - Wednesday
2020
Pokémon Direct January 9th, 2020 - Thursday
Animal Crossing: New Horizons Direct February 20th, 2020 - Thursday
Nintendo Direct Mini March 26th, 2020 - Thursday
Nintendo Direct Mini: Partner Showcase July 20th, 2020 - Monday
Nintendo Direct Mini: Partner Showcase August 26th, 2020 - Wednesday
Super Mario Bros. 35th Anniversary Direct September 3rd, 2020 - Thursday
Nintendo Direct Mini: Partner Showcase September 17th, 2020 + Monster Hunter Direct September 17th, 2020 - Thursday
Nintendo Direct Mini: Partner Showcase October 28th, 2020 - Thursday
Super Nintendo World Direct December 18th, 2020 - Friday
2021
Nintendo Direct February 17th, 2021 - Wednesday
Nintendo Direct: E3 2021 - Tuesday
Note: presentations that don't fall under the Direct brand (like the Indie Showcases / Indie World) and / or aren't aired across Europe / Americas / Asia at the same time aren't included. Pokémon Presents aren't included since The Pokémon Company handles them with more autonomy compared to the Pokémon Directs.
As you can see, outside of E3 where Nintendo constantly takes the Tuesday slot (and of course it's an event with other major parties involved), other general Directs tend to fall on the same day of the week: on Thursdays in even years, on Wednesdays in odd years. If we go by Nintendo's original planning before the pandemic really disrupted plans, the Direct Mini from March 2020 was aired on a Thursday and then we got several Partner Showcases in different days of the week (the first one aired on a freaking Monday!), but it's probably notable to see that the only Partner Showcase that was livestreamed instead of dropped out-of-nowhere happened, once again, on a Thursday. Also, starting from 2019 it looks like themed Directs were also aired in the same days as the general presentations, with the sole exception of the Super Nintendo World Direct.
2021 is an odd year, and February's Direct was indeed aired on a Wednesday. Thus, if Nintendo keeps on following their...weird patterns (seriously, why) the September Direct should be aired on a Wednesday as well.
2) when in September exactly
The February 2021 Direct was aired on February 17th, 2021. That's the date of the February 2019 Direct + 4. If Nintendo follows the same criteria, the September Direct should fall on September 8th, 2021. The first "full" September week, like the September 2019 Direct and the September 2018 one according to the original plans, before the delay caused by the earthquake. However, back in 2017, the Direct was aired in the second "full" week of September.
Also, let's look at when the Tokyo Games Show started the past few years
TGS 2017 - September 21st, 2017; 8 days after the Direct aired
TGS 2018 - September 20th, 2018; 14 days after the Direct was originally intended to be broadcasted
TGS 2019 - September 14th, 2019; 10 days after the Direct was aired
TGS 2020 started on September 23rd so one week after the Nintendo Direct Mini Partner Showcase September 2020 presentation...but 2020 was also a weird year with plans that changed a lot compared to the original intentions.
TGS 2021 will start on September 30th. Given the relatively late start of the event and considering how much Japanese third parties could try to tie their own reveals with the pre-TGS Direct, it's possible Nintendo and third parties are going with a slightly different schedule compared to 2018 (original) and 2019, that is the Direct is aired the second, if not the third "full" week of September.
3) how much the PlayStation Showcase matters for the Direct's stream date
Last year, Sony had their PS Showcase on September 16th. Several hours before the event, Nintendo announced the Nintendo Direct Mini Partner Showcase September 2020 for September 17th, 2020 and it was livestreamed not even 24 hours after the Sony event ended. They may not care that much but who knows.
...And yeah, in the end, Nintendo may say "fuck the patterns" and have the Direct on a Thursday, maybe even this week just right after the Sony event. Or they could just shadow-drop a Mini. Or it could be a Partner Showcase. Or... I mean, the latter two cases have very, very low chances of happening truth be told. But Nintendo can be both predictable and unpredictable at the same time.