No spinning cube means it’s a WebGL issue and not a Cesium issue, so there’s nothing you can do without fixing the real problem (WebGL being unavailable).
The standard procedure on the desktop:
Update video drivers to latest version
Update to latest browser version
Try all 3 browsers to see if any of them work. (Chrome/Firefox/IE11 or Safari)
If on Windows or Safari, make sure you have all OS updates applied, as they both include browser updates as well.
If the first 3 fail, the last shot is to see your video card is blacklisted by the browser (for known bugginess) and override the blacklist if desired. This is browser specific.
In this day and age getting to step 5 should be an incredibly rare occurrence. Most are fixed at step 1 or 2 unless the user has really old hardware.
For mobile/tablets, it’s tricky because you have a lot less control.
All iOS 8 devices support WebGL, so you will never get a failure there (but Safari itself is pretty buggy).
Android devices require either Chrome or Firefox but if it doesn’t work on either of them you don’t have many options. Make sure you have the latest version of Android available for the device. You can also override the blacklist if WebGL is blocked, but doing that is browser specific.