Since you have installed Ubuntu while on your desktop as Wubi installation, I believe there will be no grub installed on the desktop harddrive. Wubi modifies the vista bootloader to provide a Ubuntu boot option. It does so by modifying the boot.ini in the vista partition.
If, for example, Vista is installed in C drive, you would find C:\boot.ini which is a text file specifying the boot options. You can open that file on the host system to see how this is done on the desktop system.
Caution Wubi installs are by-design not portable across Windows installations (even between 2 machines having same version of Windows - Pls don't even try to copy the boot.ini anywhere else.). This is because of the following
- Wubi installs the entire ubuntu OS + persistent data into what appears to be a file to windows (see the 2nd item in this section of wubi faq
- Location of the mount point of the drive is not expected to consistent across all systems and hence the location of this above file as the bootloader sees it can vary from one pc to another.
Installing Ubuntu onto an external harddrive is an excellent option of being able to have your favourite os on-the-go and due to the above reasons doing it in a "Wubi-mode" kind of defeats that purpose.
My suggestion would be, if possible, to have Ubuntu on the external hard-drive as a normal install. If you could eke out some space for a new partition in the disk using gparted in the livecd/liveusb then you could follow this tutorial to even migrate your existing Wubi install into the new partition.
This link has comprehensive info on wubi that might be helpful.