This sounds like a fun but big project. Basically what is needed is code that solves the hydrogen atom Schrödinger equation in N dimensions rather than 3 (the electron and nuclear mass are free parameters). This is not fundamentally different from the standard 3D textbook case, and has already been done in various ways (see this question, this paper, this paper, this paper, and this paper points out that the potential in N dimensions should not be $1/r$, something often forgotten).
I have not seen anything about extra timelike dimensions for this case. That may make things far messier mathematically, since the demand for initial conditions go up a lot. Still, there are (I think) counterparts to the static wave-function solutions even for multiple time dimensions. This paper looks at the case where there is one time dimension for each particle.
The multi-electron atom is usually approximated as a hydrogen atom with a hopeful hand-wave that there is not too much electron-electron interaction; if this is not true one needs to use some perturbation scheme and things get complex even in three dimensions.
However, you also mentioned having different kinds of electrons. That complicates matters, since a lot depends on what kind of kinds you use. If the electrons just have some extra quantum number like spin but different (like being of different "colour") then more than two could share an orbital and the electron structure would be more packed. If the masses are also different each mass would have slightly different orbitals and the spectrum would get rather complex (since "filling up orbitals" now depends on how many electrons of each mass have been added).
In short, this sounds like a fun project. Much would be just like writing a code for finding the eigenfunctions of the hydrogen atom but with some extra quirks - there is most likely no existing program that does this. Once you have the eigenfunctions you know the orbital shapes, their eigenvalues give you the spectrum, and their degeneracy gives you the number of electrons in different orbitals. Doing the same for the nucleus is likely much more involved (given how messy it is for 3D atoms).