For TestFlight, there's nothing you can do. However, after the first upload, and Apple's had a chance to do a cursory review, the availability will go much faster.
For submissions to the App Store, there's only 1 way to make this process go (a little bit) faster and that's to request an expedited review. You would need to justify the quicker review time to Apple, although the rules are fairly lax. You can do this several times a year but after a few of them Apple might not grant them.
Here's what Apple indicates are the valid reasons for this request:
Expedited App Review
If you face extenuating circumstances, you can request the review of
your app to be expedited. These circumstances include fixing a
critical bug in your app on the App Store or releasing your app to
coincide with an event you are directly associated with.
Urgent Bug Fix
If you've submitted an update to fix a critical bug in
your app on the App Store and you are requesting an expedited review,
be sure to include the steps to reproduce the bug on the current
version of your app.
For apps associated with an event, we recommend
you plan and schedule the release of your app in iTunes Connect.
However, if your app is still in review and the launch of your event
is quickly approaching, you can request to have your app review
expedited. When submitting your request, it's important to include the
event, date of the event, and your app's association with the event.
Please Note: Expedited reviews are granted on a limited basis and we
cannot guarantee that every request will be expedited. Sign in to
request an expedited review.