[Some versions give "Anna Maria" instead of "Jeremiah". I assume that's pronounced "ma-RYE-a".]

Yes. See A Dictionary of Rhyming Slang By Julian Franklyn 1975

http://books.google.com/books?id=9zsCIMSi-dsC&pg=PA8&lpg=PA8&dq=cockney+rory+o'more&source=web&ots=1TSS-ne5mb&sig=ikSHEl_9H47QNsadoh7w9VJ3qgI&hl=en&sa=X&oi=book_result&resnum=2&ct=result#PPA32,M1>See page 32


This URL is too long for the Blicky maker, so I have had to put it in and hope