I deliberately held off from upgrading my Samsung ATIV Windows 8 Pro Atom tablet from Windows 8 to 8.1 to experience the process. It started kind of rough.
You start, not in the Windows Store, but on Windows.com.

Click the link and the upgrade process should start. Instead, I got a HTTP 403 Forbidden Access Denied error. Whoops! First day jitters. I hope someone quickly deployed more Azure instances! 😉
I tried again about 2 hours later and the link opened the Windows Store where I clicked on Download.

Then a download starts. For me, on a fast line, it was slow.

After a long wait I was informed that the machine would reboot automatically after a 15 minute wait. I triggered it to happen immediately. The very slow count from 1 to 100% began after the reboot. Bear in mind that the storage in these tablets appears to be very slow.
My machine hung on a black screen during this stage and that stuffed the install. I rebooted and it reset this stage of the upgrade. Another power reset and the issue repeated.
And then I was greeted by
Restoring Your Previous Version Of Windows
Yes, the upgrade failed for no reason. So much for the upgrade process. Looks like I’ll be doing a fresh install instead.
FYI, I saw another machine (also Windows 8 OEM but another hardware brand) even refuse to upgrade for an unknown reason. This is what I was greeted with when Windows 8 was restored and I logged in.
Couldn’t Upgrade To Windows 8.1
Sorry, we couldn’t complete the update to Windows 8.1. We’ve restored your previous version of Windows to this PC.
0xC1900101 – 0x40017
This is why there is value in public beta testing, unlike in the much smaller restricted private TAPs.