When I click "Download" in the High Sierra page in Mac App Store, the button turns into a gray "Downloading" button and might even show an estimated download time but after a while it turns into a gray "Download" button that isn't clickable anymore.
This happens on my 2 Mac's: a Mac Mini running Sierra and a MacBook Pro (13-inch, Mid 2012) running High Sierra Beta (17A360a).
Update: I've just restarted the download and it estimates about 1 hour to download the ~5GB of the installer. If I start another simultaneous download in the network (iOS update, for example), the transfer time starts to increase and if it reaches some value over 15h the App Store quits the download. This happens even if the installer download is still active, only at a slower rate, as can be verified by looking for "OS update" in the Console app.
Update 2: it seems the problem I was facing with the download “fixed itself”. Since yesterday I managed 2 full downloads without any issues on my 2 Macs.
Best Answer
You can't update because your system is running with a higher beta Version of OS X. Leave Apple beta Program with the devices and wait for a greater update than the system have at the moment.
If you need help leaving the beta Program check this: https://beta.apple.com/sp/betaprogram/unenroll