Seeing your template, in my humble opinion, it seems to use the fourier package.
Here you can find several fonts with LaTeX math support: https://tug.dk/FontCatalogue/mathfonts.html
From the context I think it is clear that it means (what we would now call) a stripped down distribution of tex rather than an implementation of tex.
Note that your statement
Considering the fact that plain.tex cannot be changed
isn't accurate: plain.tex says you can change this bit of the file
% (And those that were \preloaded above can be ...
Here's a babel implementation showing English (australian) as main language, simplified and traditional Chinese, and Pinyin.
It will compile with XeLaTeX and LuaLaTeX. Though be aware that the Chinese fonts are huge and it takes the LuaTeX font loader a long time and a lot of RAM to build the cache. So long, that I can't actually compile this on my computer ...
I’m not sure I understand your question properly. If you need to use two different calligraphic alphabets in the same document, the easiest way to accomplish this is to load one as \mathcal and the other as \mathscr in mathalpha. (On older installations, you might need to use its older name, mathalfa.)
This is not exactly with ucharclasses, but babel supports the Sinhala language with font families. It requires some tagging, but you appear to be using a \sinhala command anyway. Keep using XeLaTeX (it might also work in HarfTeX.)
% To fit within a TeX.SX MWE:
I’m using the Monotype Library Subscription via SkyFonts on macOS (10.14.5). fontspec is able to access the fonts provided by SkyFonts if you address them by their PostScript name. For instance, I have the “FF Meta Pro” family installed and can use it with XeLaTeX as follows: