I don't know how much of this is necessary, but doing all of it worked for me.
If you try to install SP1, it will do the install, reset, & fail to update the registry. It then automatically reverts to the previous setup, uninstalling the SP.
The error code returned will lead you to an article on the MS site, & a download for a file, hotfix for windows (KB947821) with a similar filename to the SP (I don't know the actual filename, as I renamed it "run 1st" so I'd know). MS will tell you that the problem is known, & is due to a problem in the store, used to hold temp files, or something.
If you execute this then run SP1, it still may not work. I got an error:
ERROR_SHARING_VIOLATION(0X80070020).
MS themselves don't have an answer to this, other than it may help to disable your antivirus (it didn't). After much digging, I found the following advice. This worked for me, though I don't know which combination, I just did it all to make sure.
Go to:
Control Panel>Programs & Features>Windows Features
Uninstall MS .NET 3.5.1
Back in Programs & features:
uninstall Avast (I assume this is also necessary for other AVs. Disabling it didn't work for me).
Disable Windows Firewall.
Disable Windows Defender.
Install hotfix.
Install service pack.
Reinstall .Net, AV & enable firewall & defender.
Halelujah!!