It’s hard to say without specifics but most BLAS/LAPACK implementations today aren’t written in Fortran, they just provide a Fortran interface - it’s worth noting that BLAS/LAPACK original implementations were written in the 1970s in Fortran but they are not generally used today because other libraries with the same interface have been written that provide better performance. The open source implementations OpenBLAS and Atlas are written in C and commercial cuBLAS, rocBLAS and MKL are C++. This is also true for many other OSS scientific libraries I’ve used - Sundials, FFTW, etc.
In terms of column vs row major order - the underlying implementation will have its own default way of doing it (depends on the implementation) but you can if necessary recast matrix operations in such a way as to avoid actually doing transposes in the interface layer with the other language. so there’s not really a performance penalty as no additional work is done. CuBLAS and MKL’s BLAS implementation even let you pass parameters in about your input data’s memory layout.