Thank you for the replies, Roger and Donie,
I really don't know much else about Jeremiah Sullivan but do appreciate any feedback or tips for searching further.
Just as a matter of interest, I am guessing that he married Catherine (surname unknown) in England because the census records state that she was born there, and so were their children.
Their son Matthew born c1834, of our direct line, also married a lady from Ireland, Mary Flury, spelling as per their Roman Catholic Church marriage record in London. However, the hand-written birth record for their son, Matthew born1860, which i obtained from the General Register Office in England, shows his mother's name as Mary Sullivan formerly Floyer, which seems to be a surname in it's own right and not necessarily just a variation of Flury. Her mother appears as Mary J Floury, born in Ireland, in one of their census documents. So three spellings and again no real luck with tracing the Irish heritage.
Also, there are not only a lot of Jeremiah Sullivans born in Ireland, and even specifically in Cork, around his time, as you have confirmed, but also quite a few Irish Jeremiah Sullivans of that age in England around his time, and even specifically in his area around Middlesex. Also quite a few military record listings, but all with no information to help.
I will follow both of your advice on the way forward, however, I do not know what parish he comes from because the only clue is the 1851 census document entry which states that he was born in "Ireland, Cork".
With best regards,