I had the same problem on Win 7 and this worked for me in command prompt.
Solution 1:
RD/S pathname
example:
RD/s C:\Users\Username\Downloads\Music
this will delete all files and sub-folders in the folder, including the root folder itself.
Solution 2:
Run cmd, navigate to C:\ (or other disk, if you have installed it in different path).
step 1:
takeown /r /d y /f cygwin //This command takes ownership recursive of the folder, without asking anything
step 2:
icacls cygwin /t /grant Everyone:F //This command gives Full Access to Everyone recursively in the folder
step 3:
rmdir /s /q cygwin //And finally, the command which deletes it all and removes Cygwin
reference:
http://problemsolv.in/2012/09/how-to-remove-cygwin-permission-denied-problem/