When the Windows 10 upgrade fails with "Operation failed: Mount WIM file"

I continued attempting to upgrade a 64-bit Windows 8.1 laptop to Windows 10 today. It would consistently get to about 24% in the pre-reboot setup screen (when the old OS was still running) and then fail with the generic "Something happened" message box. This appeared at the end of the setuperr.log file:

2016-06-09 11:37:53, Error                 SP     CMountWIM::DoExecute: Failed to mount WIM file C:\$WINDOWS.~BT\Sources\SafeOS\winre.wim. Error 0xC1420127[gle=0xc1420127]
2016-06-09 11:37:54, Error                 SP     Operation failed: Mount WIM file C:\$WINDOWS.~BT\Sources\SafeOS\winre.wim, index 1 to C:\$WINDOWS.~BT\Sources\SafeOS\SafeOS.Mount. Error: 0xC1420127[gle=0x000000b7]
2016-06-09 11:37:54, Error                 MOUPG  MoSetupPlatform: ExecuteCurrentOperations reported failure!
2016-06-09 11:37:54, Error                 MOUPG  MoSetupPlatform: Using action error code: [0xC1420127]
2016-06-09 11:37:54, Error                 MOUPG  CDlpActionImageDeploy::ExecuteRoutine(329): Result = 0xC1420127
2016-06-09 11:37:55, Error                 MOUPG  CDlpActionImpl > > >::Execute(441): Result = 0xC1420127
2016-06-09 11:37:55, Error                 MOUPG  CDlpTask::ExecuteAction(3243): Result = 0xC1420127
2016-06-09 11:37:55, Error                 MOUPG  CDlpTask::ExecuteActions(3397): Result = 0xC1420127
2016-06-09 11:37:55, Error                 MOUPG  CDlpTask::Execute(1631): Result = 0xC1420127
2016-06-09 11:37:55, Error                 MOUPG  CSetupManager::ExecuteTask(2024): Result = 0xC1420127[gle=0x0000013d]
2016-06-09 11:37:55, Error                 MOUPG  CSetupManager::ExecuteTask(1987): Result = 0xC1420127[gle=0x0000013d]
2016-06-09 11:37:55, Error                 MOUPG  CSetupManager::ExecuteInstallMode(804): Result = 0xC1420127[gle=0x0000013d]
2016-06-09 11:37:55, Error                 MOUPG  CSetupManager::ExecuteDownlevelMode(389): Result = 0xC1420127[gle=0x0000013d]
2016-06-09 11:37:58, Error                 SP     CDeploymentBase::CleanupMounts: Unable to unmount the directory C:\$WINDOWS.~BT\Sources\SafeOS\SafeOS.Mount. Error: 0xC142011C[gle=0xc142011c]
2016-06-09 11:38:12, Error                 MOUPG  CSetupManager::Execute(232): Result = 0xC1420127
2016-06-09 11:38:12, Error                 MOUPG  CSetupHost::Execute(371): Result = 0xC1420127

The mention of WIM mounting made one user in the Super User chatroom recall the WIMMount Registry entry. I removed the subkeys of this:

HLKM\SOFTWARE\Microsoft\WIMMount\Mounted Images

After I did that, setup ran successfully to completion! (I had also run dism /online /cleanup-image /restorehealth along the way, but I'm not sure that did anything relevant.) That laptop is now running Windows 10.