An English translation of "Die Klassenkörper der Komplexen Multiplikation" was posted on arXiv by CheeWhye Chin in 2020: https://arxiv.org/abs/2006.04531
English translation of the proofs of many theorems from "Die Typen der Multiplikatorenringe elliptischer Funktionenkörper" can be found in
- chapter 14 (eg. Theorem 14.18) of Cox's Primes of the form $x^2+ny^2$ (2nd edition, 2013),
- chapter 13 of Lang's Elliptic Functions (2nd edition, 1987), and
- chapter II (section 4 and 5) of Silverman's Advanced Topics in the Arithmetic of Elliptic Curves (1st edition, 1994).
- Historical discussion about this paper is available in section 8.4 of Roquette's The Riemann Hypothesis in Characteristic p in Historical Perspective (1st edition, 2018)