The paper Conformal mapping for the efficient MFS solution of Dirichlet boundary value problems by Andreas Karageorghis and Yiorgos-Sokratis Smyrlis (Computing, May 2008, Volume 83, Issue 1, pp 1-24) seems to provide the formula you desire (section 2.2), as well as more references apropos of its derivation.
The paper On conformal representations of the interior of an ellipse by Kanas & Sugawa seems to meander around your question. In the first section they remark that Schwarz found such a map back in the day, but so far I have been unable to locate it a source for it. Section 3 discusses an ODE which one of your desired functions satisfies.
The paper Ellipses, near ellipses, and harmonic Möbius transformations by Martin Chuaqui, Peter Duren and Brad Osgood (Proc. Amer. Math. Soc. 133 (2005), 2705-2710) deals with mapping the boundaries of the domains you're considering. While it is not immediately useful, it may be of interest. At the end, the authors remark that a conformal map of the form you desire will ''not be able to map concentric circles to concentric ellipses''.