"Union member" would translate as "συνδικαλισμένος" or "μέλος συνδικάτου" (or possibly "μέλος του συνδικάτου" if referring to a union already mentioned).
As for "unionized work", in our times this would rather refer to the mere existence of a union: "στη δουλειά μου υπάρχει συνδικάτο" (or "σωματείο"). In older times, it might entail connotations of closed shop, obligatory contributions by workers to the union, or control of hiring by the union, which would require a rather descriptive translation.