Whereas the German word homogen is rather straight forward to pronounce, the English equivalent, homogeneous, is slightly trickier. Oxford English suggests [ˌhɒməˈdʒiːniəs] (British English) and [ˌhoʊməˈdʒiːniəs] (American English), see and listen here. On the other hand, the Longman Pronunciation Dictionary includes the following possibilities:
[ˌhɒməʊˈdʒiːniəs] (used by 75% of the speakers)
[ˌhoʊməʊˈdʒiːniəs] (used by 25% of the speakers)
Hence, while the pronunciation of homo- is not unique, all these possibilities are identical regarding the pronunciation of -eous. (If you are unsure about the subtle difference between [əʊ] and [oʊ] listen to the pronunciation of flow.
A common mistake, possibly related to the alternative but outdated spelling homogenous, is to miss the second letter e and pronounce “-eous” as [əs], as for example in the word dangerous. German speakers in particular should try to avoid the false friend homogen (rather common in written English, not so common in spoken English).
Even if you know how to pronounce homogeneous, the pronunciation of homogeneity can be rather difficult (the German word Homogenität is not simple either but at least the spelling is phonetic). According to Oxford English, the correct pronunciation is [ˌhɒmədʒəˈniːəti] (British) and [ˌhɑːmədʒəˈniːəti] (American), listen here. According to Longman’s Pronunciation Dictionary, the same variations of homo- occur as in the case of homogeneous. Moreover, it is not common practise but acceptable to pronounce “-eity” either as [eɪti] or simply as [iti].
Finally, let me mention that the above also applies to the words inhomogeneous and inhomogeneity, as well as heterogeneous and heterogeneity.