You may not be safe on a new computer. Check your licensing terms for Lightroom. You may only be licensed for one computer. You may be able to transfer the license to another computer, but that could mean another session with Adobe tech support. I wouldn't count on it being a simple installation on a different computer.
I've run into this issue using Adobe Photoshop Elements. It used to be a one time purchase, but I couldn't run it on a different computer if I'd already installed it on one. When you go to validate the installation it will say that it's already been validated on a different computer, so it has to be removed from the old one first. It will likely be problematic with older versions, since Adobe won't make the process easy. Like everything, they want to frustrate you into buying the latest version.