While I agree that we may have to wait for a more specific placement in doomsday and work backwards not all the trees were edited to look like fall, I believe the trees outside of Red Guardian's house in the trailer were a nice green. So if dependant on the US Location some trees are green and some aren't (and going off of filming locations) then I would be pushing for Late Sep/Early Octover.
I also personally dont think this will be so tight with DD & BNW allowing time for whatever DD S2 is going to do and NYC to allow the watchtowers development. I also think Bucky's congressman timeline is just straight up messy.
John Walker's flashback is set "a few short years" after his unveiling, confirming at least 2027 so I will place that flashback within 2-3 months prior to the movie since it seems pretty ~fresh~ to Walker. I will also be placing it after BNW as it would be fitting for new Cap to do a very public save and the disgraced Cap to have an article written about him.
It terms of the desert scenes while yes it is the desert and things are brown and sandy, it was filmed in Utah and grasses still change in different seasons. A majority of natural grasses there do still brown out completely in the late summer / early fall as is shown in the movie.

while there is a chance this is just hay placed by the production team, they could have made it look like spring (like they did for BNW) if they wanted to.
Not a very solid point but NYC is usually quite sunny in the late spring & summer and this movie was very cloudy there. Now they can still have cloudy days other days but also seemed like the vibe of the movie.

I know the black is creeping in this photo but in this scene definitely wasn't peak fall IMO.
Ill be placing pseudo flashbacks when it is released digitally since the void flashback scenes are somewhat a flashback to real events (but won't be cutting it out of the present movie).
My final consensus
for now is Late Sep 2027.