Time Machine alternatives to save registration information When restoring a Time Machine back-up to a new machine, none of the registration information for the purchased programs carries over to the new machine. 

Is there a back-up program which would contain this information? I've got dozens of programs and in some cases the developers have disappeared. 

Looked at this similar question: Time Machine alternatives to clone one mac book to another

but it doesn't contain any information about restoration of registration information for purchased programs.
Time Machine backs up every file on the target volume. So any method of storing the registration that involves files on your disk can be restored from a TM backup. No other software is required.

So it's up to you to restore the required files. For instance, relevant data may be in .plist files in the user or root level Library/Preferences or /Library/Application Support folders.

However, many developers understandably use hidden files and obfuscated methods to store registration details and confirm authorised usage. 

If you selectively restored files and folders from your backup, then you won't have restored those hidden files. You can see hidden files in the Finder and Time Machine by pressing Shift Command . 

It may be easiest just to clone the entire disk, so you may want to try SuperDuper! or Carbon Copy Cloner; though I've found that this doesn't always work: Devs may hash hardware characteristics, such as the volume's UUID, in order to ensure that the software won't work without re-registration on new hardware.

It's always a good idea to keep all the serial numbers in text files in a folder for just this occasion, which isn't usually needed very often. I also still have the emails which provided the serials when I made the purchase.