Dick Justice has 'the tent', but Jordan has 'elephant' in that line - he pronounces it like Johnny Jones does in his version of 'The Dirty Dozens': 'air-phant'. I will stick with 'ugly' in Jordan's version. I thought it was 'earthly' at first (which made little sense), but then I came across the stanza in Oliver's book. Once I knew what it was, I could hear it clearly enough. It certainly sounds nothing like 'nursery' in Jordan's recording.
'Whiffin'' would certainly make more sense. In both the Jordan and Justice recordings, it could well be 'whiffin''. Justice definitely sings 'I'm simply wild about my good cocaine', but this could be the result of a mishearing of the original.
Your 'cradle' theory is interesting, but I reckon it falls down because both Jordan and Justice very clearly sing 'THE baby's in the cradle' not 'baby's' nor 'my baby's'. There is no 'doctor' in the line in Jordan's version: 'Now, the baby's in the cradle in New Orleans, he kept whiffin' till it got so mean'. Justice has 'doctor' in his line: 'Now, the baby's in the cradle in New Orleans, the doctor kept whiffin' till the baby got mean'. When I first listened to Justice's version, I misheard 'dog' - probably an association with 'whippin''. In a cover of Justice's version, I have heard the following variation:
Baby in the cradle in New Orleans Doctor kept a-whippin till the baby got mean Then they stuffed up his sniffer till the baby got sore The doctor said the baby couldn't smell no more