So... 24 hours later I found the solution. Apparently this behavior (Access denied to stop/start any win services and the consequent System error 5 has occurred sample here) is native to all newer WIN OS. As I found out, there seems to be no straight forward way to elevate the user privileges directly from CMD when executing .bat file so every command which follows (i.e.: stop or start service) will fail due to a lack of user privileges.
My solution
1) Download the Elevation PowerToys 2.0 (this is just an executable which creates a folder "Elevation" with several files)
2) from the freshly created "Elevation" folder double-click InstallAllPowerToys.cmd file (this actually installs the additional options for you to use the word "elevate" in the CMD to run any .bat file directly as an administrator (without being bugged by win)
3) go to your Control Panel -> User Accounts -> Change User Account Control Settings -> ..bring the slider all the way down to "never notify" level. (this ensures the suppression of the "run as administrator" popup boxes)
4) create a wrapper .bat file which will be executed with elevated privileges i.e this is a bat file which I called backup.bat and its content is as follows (note my website folder in this case is located on W:\ drive):
elevate cmd /k "w:\mywebsite\antbackup.bat"
5) create another .bat file with the actual Apache Ant call to bild.xml file (with all the details - which services should stop or start and where should be my files Zipped and backed-up, etc...). This file could be called antbackup.bat:
ant -buildfile w:\mywebsite\build.xml antbackup
pause
6) double-click the backup.bat file (of have your scheduled tasks do it, cross your fingers and hope for good results :)
This worked for me. It could help someone else as well.