I'm not sure how to get there, but maybe I can help characterize fast play. Let me show you a simple position I've won many times in speed games. It's interesting to me how players of different skills react.
Click forward to black moves Ne4.
Weaker players, perhaps pigeonholed by the previous threat on the rook, seem blind to the king side threat (f2 is mate no less!). I often see moves like Bb2.
Other players apparently blunder check the Bb2 move and notice f2 is mate. So after a long-ish thought by them I'll see a move like Qe2.
I can tell you in similar situations (if I were white facing a new position) the c3 and f2 square would "light up" right away. My first instinct would be to "touch" those two squares and I'd find Qf3 quickly. What's interesting is when players play Qf3 they find it much faster than those who decide on playing Qe2. Why? I think the Qe2 player is disorganized in their threat recognition. The process is slow and cumbersome to them. Pieces aren't "reaching out" and "touching" squares / squares aren't "lighting up" or however you want to say it.
-----------------------
My best guess for advice would be to play lots of blitz and always ask the question "what's the threat?" You'll lose a lot so it will force you to quickly identify threats. That combined with solving many tactical puzzles daily I would guess should do the trick. It's not so much about thinking as it is conditioning yourself to automatic actions / responses.
I have a huge problem with timed chess
if you look an my online chess rankings i am in the 1600s but in the standard chess my rating falls all the way down to the 900's and in bitz in the 500s its logical that there should be a fall but not that steep a fall. In blitz chess i miss things that even a beginer should see right off the bat.
This has really kind of been the story of my life in undergrad I would get C's and D's on tests dispite studying hard but my projects and papers would all be A's. However i found new ways of thinking and studying starting getting B's on my tests.
How can I help my speed and quick thinking in chess besideds just study harder.