Apparently the 1803 update is now installing on Alienware laptops. After one failed try at updating this afternoon, I contacted Microsoft Support and they recommended that I clear the Windows Update Cache, then try the update again. The second attempt downloaded a new 1803 update and installed perfectly on my Alienware 17 R5.
Microsoft’s recommended procedure for clearing the Windows Update Cache:
Press Win + R and then type services.msc and hit Enter.
Find and double click on Windows Update and then click on Stop button.
To delete Update cache, go to - C:\Windows\SoftwareDistribution\Download folder.
Press CTRL+A and press Delete to remove all files and folders.
Again press Win + R and then type services.msc and hit Enter.
Find and double click on Windows Update and then click on Start button.