>>Are there any potential problems I might face when upgrading the OS of an existing SQL Server install?

Quick update in case anyone comes across this in the future.

The integration services service wouldn't start and neither would SSMS load.

The relevant error seemed to be “The file C:\WINDOWS\Microsoft.NET\Framework\%WTF%\mscorlib.tlb could not be loaded.”

I needed to repair the .net installation by going to C:\Windows\Installer\3e60d.msi, right clicking on the file and choosing repair.