Time for an update - and a workaround.
I just tried the latest 5.2.4.2 release - 64-bit - and I still have the same problem: every application hangs on startup after displaying the LibreOffice logo. (Just stating this once more for those who jump to conclusions: it’s not an OpenGL issue.)
So, I tried the 32-bit version and guess what? It works without any problem! The workaround is to use 32-bit LibreOffice and avoid the 64-bit version for the time being - if you have this problem.
It seems to be too trivial to blame 64-bit code for the problem, but - as others have pointed out - the 32-bit version works while the 64-bit version hangs on application startup, on some machines at least. Just to recap, I did a number of clean installs of the 64-bit version and disabled OpenGL, so this is not a registry/configuration/OpenGL issue.
As this is an unsatisfactory conclusion, I have also raised bug report 105084: