That's a difficult one. The English and German concepts of labour and contractions are different, as far as I know. Moreover, neither English nor German terminology is strictly defined in that area.

You are probably on the safe side, if you use Braxton-Hicks contractions for "Schwangerschaftswehen".
But "Vorwehen", especially if they are to occur before "Senkwehen", are really the same as Schwangerschaftswehen in German, and a translation like pre-labour, prodromal contractions or false labour may be misleading, because these terms are mostly used for intensifying Braxton-Hicks contractions during the days before "true" labour starts which then leads to effacement and dilation of the cervix. You could work around by using any of the (equally ill-defined) synonyms like "practice contractions".

Senkwehen occur in the last month of pregnancy and lead to dropping of the baby. I am not aware of a specific term for this in English; they are included in the Braxton-Hicks contractions, as far as I know.

Geburtswehen is not really a technical term, more often used in a figurative sense, or as a general term for any labour during birth. The first stage of labour (as suggested by the sequence of your terms) would be "Eröffnungswehen".

Presswehen is the second stage of labour (one obvious correspondence at least, in your list).

Nachwehen is sometimes (infrequently) used for "Nachgeburtswehen", the third stage of labour expelling the placenta. More often, Nachwehen describes afterbirth pains/afterpains occurring in the days following delivery.