Here’s what I know about compiling lapack from GitHub - Reference-LAPACK/lapack: LAPACK development repository.
Classic flang can compile and execute lapack. https://github.com/flang-compiler/flang
New flang can’t execute lapack yet; its code generator is still being written. https://github.com/llvm/llvm-project.
The code generator is being developed downstream of llvm-project in the fir-dev branch of https://github.com/flang-compiler/f18-llvm-project. That branch can compile lapack but not yet compile all of the included test programs.
I expect that fir-dev will be able to execute the lapack test programs in a few months. There’s an on-going project to upstream fir-dev to llvm-project; it’s hard to predict when all of the required code will be available in https://github.com/llvm/llvm-project.