Die Abkürzung im Deutschen war mir auch geläufig. Habe mir dann überlegt, ob die Engländer das nicht auch abkürzen. Scheint der Fall zu sein.
While "CW" may indeed exist, in AE, it is not terribly common and shouldn't be used (unless it is in internal communication of a company where you know that the folks on the west side of the Atlantic know what it means.
As a reinforcement of what has already been said, even if the concept of "calendar week" exists, how it is typically understood and used in AE at least differs from how it is used in German. Following through on the link above: Noun 1. calendar week - a period of seven consecutive days starting on Sunday
That reflects the older and AE practice of viewing that the first day of the week is Sunday.
Also, to the degree that weeks are numbered in AE, the system used to determine which is week 1 differs slightly than the ISO standard.
Thus, for clarity and wherever possible, follow that suggestion in #2