I finally managed to track down the answer to my question, after finding a reference to it in Kenneth Williams' diaries. The poem is by Stanley Baxter, whom Kenneth Williams knew in the second World War when they were both performing in the Combined Forces Entertainment unit. The poem is actually entitled 'Berlin in the _Twenties_ ', not the Thirties. As far as I'm aware it has never been published. I believe that it was first broadcast in the one BBC Radio 4 programme referenced in Williams' diaries - on 14th February, 1970 - in which Williams read a varied selection of poetry. I don't know of any other broadcast.
Due to the kind help of Kenneth Williams' biographer I have now seen the poem (as transcribed by Williams himself, he thinks) but for copyright reasons am not able to reproduce it.