Unfortunately, most applications installed on Windows are not "portable" - i.e., after installing them on one operating system, you can't just boot another operating system and run the installed program.

The reason is that the application is not only the program file on the drive - it is also its "registry" settings, it may install libraries into the operation system itself, maybe some configuration and so on. This is specifically true for applications like a dictation software that needs to hook into all kinds of dark corners of the operating system to support "typing" into other applications.

Dragon Naturally Speaking can - theoretically¹ - be run under Linux using the Wine compatibility wrapper - see the Dragon Naturally Speaking WineHQ AppDB entry for details - as some versions work better than others, but you'd need to actually install another copy of Dragon into your Linux operating system using Wine, and for that you'd need access to the license details.

I'm not familiar with the specific program under which you have received a Dragon Naturally Speaking license, but if under that program you are entitled to a license key and not just a single installation, then you should be able to install two copies under the same license - as it is the same computer.

¹ I have never tried this myself.

If your laptop is powerful, say with a quad-core processor and plenty of RAM, then you might consider running one OS or the other in a virtual machine. (I don't know about other virtualization software, but in VirtualBox the user designates how many processor cores and how much RAM the virtual machine gets, and when the virtual machine is running the host OS doesn't have the use of those resources.) You could copy-and-paste text from Naturally Speaking to a program running in Ubuntu, although that process would be somewhat clumsy, because extra keystrokes would be required to move from the host OS to the client OS, and vice-versa. Still, it could be done; if I were you, I'd want to dictate a paragraph or two before copying to the other OS, rather than copying one sentence at a time.

Assuming that your laptop is sufficiently powerful, it wouldn't be too difficult to try installing virtualization software in Windows and then create a new installation of Ubuntu in a virtual machine. You could keep the file for the virtual hard disk in your external hard drive. There is a bit of a learning curve involved when running virtualization software, but I personally found it to be very worthwhile. (Like I found experimenting with Ubuntu to be worthwhile, ha ha.)

There is an interesting Wikipedia article about speech recognition software for Linux here, which discusses several of these points.

EDIT: If you can make Simply Speaking work through Wine, then that would be a great way to go. It can be a real pain to make programs work with Wine though... There is also a commercial product, CrossOver Linux, that seems to be a fancy version of Wine, or a better way to set up Windows programs in Wine. That might be worth checking out. Good luck!