Well, first - realize that there's little point to this: Once you've taken into account the reduced-mass correction, the next correction (by order of size) would be taking into account relativistic effects, first the relativistic momentum of the electron, then spin-orbit coupling, the Darwin term, the Breit interaction, and finally vacuum fluctuations/QFT effects. Then the finite size of the nucleus comes into play! (at which point you're down at parts-per-million or so of the energy) So it's probably hard to find someone who's done this calculation, since with the Schrödinger equation, it's not going to give you any more accuracy.
But purely academically it might be fun to try to do.
I doubt there's an analytical solution. But if you just want the ground state, you could try a kind of simplified Hartree-Fock variational approach;
Ignore correlation and write your wave function as a product of nuclear and electronic functions. Write your hamiltonian (Tel + Tnuc + V) in polar coords. Using some set of basis functions, start with the ordinary hydrogen solution as a starting guess, use it to calculate the nuclear wave function, now minimize the electronic part and iterate until (hopefully) you reach self-consistency. (Although I don't feel convinced you would.. but I'm a bit too tired to think about it now. I may be wrong for some very obvious reason)
You'd probably need a large basis set since the effect is so small though. It's an idea anyhow.
