if you are getting this error it means that no public beta is avilible at that moment. FOR IOS 10 WE MUST WAIT TILL CLOSER TO THE END OF THE MONTH
You keep saying this, but it has nothing to do with the public beta.
The "Software Update Failed" message appears for developers attempting to do OTA updates on some devices. But the OTA updates do work on many devices. The iPadPro 9.7" gives this error, after the user accepts the terms and conditions, there is a download attempt, and as it it about to complete the box appears.
The full message is "Software Update Failed - an error occurred downloading iOS 10 Developer Beta 2."
I had the same issue with my iPad Pro I unboxed with iOS 9.3. I made several attempt to install iOS 10 Public Beta 1 over the air after installing the profile which also kept showing me a black screen during restart only to return to normal. I spent two days trying hard to avoid wiping the progress I've made with my Apps and music as curated. Just before I got really frustrated, I decided to download Opera VPN as recommended by a Developer friend of mine since I wasn't in the States. I did this and got locked on US server, then I connected the device (iPad Pro) to iTunes and backed up the device and archieved same (under iTunes Preferences). I then proceeded to update to iOS 9.3.2 which downloaded smoothly and updated without a hitch. I then ran through the normal process of the iOS 10 Public Beta 1 by enrolling the device again, installed the profile and downloaded the iOS Public Beta 1 OTA (though I had to do this severally with errors) before it finally finished and prompted me to install and there ended my frustration.