Tried every way to update from Win 8.1 to Win 10 Home and every time the dreaded 80070003 error no matter what I tried. Total frustration and almost given up after 2 weeks of trying.
Then I stumbled across the thread below, did exactly what it said and Bingo! Update worked perfectly.
The problem stemmed from partitioning my hard drive with EaseUS Partition Manager.
See link below for details and solutions if you have EaseUS installed or have used it on your computer.
Solved Windows 8.1 Recovery - Could not find recovery environment - Page 3
Hope it works for you.