As stated in the subject of this e-mail, the command MathematicaScript became wolframscript in the newer versions of mathematica. To make ob-mathematica.el work again with newer versions of mathematica simply replace all occurences of MathematicaScript with wolframscript in that file.
An extra comment. Wolfram made it's Wolfram Engine freely available <https://writings.stephenwolfram.com/2019/05/launching-today-free-wolfram-engine-for-developers/> for developers. And I guess it can be used to, for example, produce Mathematica Plots (or graphics) in the jupyter notebooks using the Wolfram Language kernel for jupyter <https://github.com/WolframResearch/WolframLanguageForJupyter> (I am not sure because I have a licensed Mathematica version and I guess my jupyter is using that kernel to get the graphics). Would be great to have such functionality working in org-mode (with-babel). Unfortunately I am not a developer. But someone else might like the idea too.