Indeed this dialog is only onscreen until another key is pressed. Therefore, it's impossible to get to this dialog except by waiting for it to collapse then getting to it inside the Omnibox. 

The following revision refers to this bug:
  https://chromium.googlesource.com/chromium/src.git/+/c4cb73ecadad6fcb2536101a227c55e114017e5e

commit c4cb73ecadad6fcb2536101a227c55e114017e5e
Author: James Wallace-Lee <jamwalla@chromium.org>
Date: Fri Sep 07 21:42:36 2018

ZoomBubbleView accessibility fixes

Don't close the ZoomBubbleView on timeout if one of the buttons inside
it has focus. (The bubble doesn't close on timeout if the mouse is
inside the bubble; screenreaders should have the same experience).

Also add an accessible title (read with Insert+T on JAWS and NVDA).

Bug:  849486 
Test: ZoomBubbleBrowserTest.FocusPreventsClose

Change-Id: Idbe2e46ab0c6d896691850ff6fee5a843abee2f9
Reviewed-on: https://chromium-review.googlesource.com/1189044
Reviewed-by: Evan Stade <estade@chromium.org>
Commit-Queue: James Wallace-Lee <jamwalla@chromium.org>
Cr-Commit-Position: refs/heads/master@{#589673}
[modify] https://crrev.com/c4cb73ecadad6fcb2536101a227c55e114017e5e/chrome/browser/ui/views/location_bar/zoom_bubble_view.cc
[modify] https://crrev.com/c4cb73ecadad6fcb2536101a227c55e114017e5e/chrome/browser/ui/views/location_bar/zoom_bubble_view.h
[modify] https://crrev.com/c4cb73ecadad6fcb2536101a227c55e114017e5e/chrome/browser/ui/views/location_bar/zoom_bubble_view_browsertest.cc
[modify] https://crrev.com/c4cb73ecadad6fcb2536101a227c55e114017e5e/chrome/browser/ui/views/page_action/page_action_icon_view.h

+a11y testers, the above CL made some changes to initial focus and auto-timeout. The initial timeout is still relatively quick, I need to press Alt+Shift+A before the screenreader finishes telling me that shortcut. Can you take another look (the fix is in canary) and let me know if it's improved enough to mark this bug fixed? Thanks!

I'd like to close this since the specific bug is technically fixed, but I think the keyboard experience could still be smoother. Especially the quick timeout, though I don't know what a better solution would be.

+collinbaker just FYI, thanks for taking on some related zoom bubble work!

Chrome: 72.0.3604.0 (Official Build) canary (64-bit) (cohort: Clang-64)
I was only able to focus the zoom button with alt+shift+a on first NTP open from the omnibox.

From content, nothing happens.
From toolbar or bookmarks bar, nothing happens

Created  issue #920005  for the ALT-SHIFT-A problem, as it's a separate issue.