I use the "General-Plugin-Template-Project" (GPTP) for my mod. GPTP has 2 versions. One for Visual Studio 2008 (older), one for Visual Studio 2019 (newer).
I used to work with the old version and VS2008. Now I am moving to the new version and VS2019 because it allows more modifications.

Is there an elegant way to move all the changes that I made to the old version of GPTP into the new one, or will I have to copy over the code blocks "by hand"?

It depends how you're programming.
If it were me trying to port code from the old version to the new, I think I would use WinMerge (or another diff files system) to see a comparison of base 2008 files and modded 2008 files, then copy the blocks that are marked as changed in the 2019 corresponding code, fine-tuning it as necessary.

Note that it is also possible to take most of the code of the 2019 version to bring it to the 2008 (there's no code incompatible with a specific version of visual studio), the only real issue are the project files (need to attach new files, detach files that got removed, deal with files that moved around).

The biggest complication may be with code/files that moved around.

Well, I now carried over everything "by hand". It wasn't as much of a chore as I had feared, though it took me an evening. Luckily, the new code is very similar to the old one. Most of my "own" codeblocks I could copy over unchanged. Thank you!