I was originally looking for a conformal map that maps a punctured unit disc to unit disc. The only answer I can find lead to this resource.
The final step of the answer given rely on a conformal map that maps an ellipse to a unit disc. Although we know such a map exist by Riemann Mapping Theorem, is there any way to write down the map explicitly (let's say, length of axes are given)?
The only related formula I can find is Joukowski transformation which does the other way around.
The Joukowski map produces a slit ellipse, and is of no help here.
For a full ellipse there is no simple formula, but an interesting procedure, using the so-called Bergman kernel. See
Peter Henrici, Applied and computational complex analysis, Volume 3, Wiley 1986, pp. 529–552. The case of the ellipse is treated on p. 546 and p. 550.