IVORY/PUDDEPHATT, Hemel Hempstead, 18th century
You don't mention your sources but if it was the IGI on familysearch I suggest that you read "The Limits of Familysearch" as the majority of the entries relating to Joshua are not directly from the parish register. It is important that you check entries against the parish register microfilms at you local LDS Family History Centre. It is also not clear whether you have any evidence, apart from the co-incidence of name, which makes the link between the Joshua Ivory born in Hemel Hempstead and the one married in St Martin in the Fields. (see "Right Name, Wrong Body".)
I haven't looked at the original registers but familysearch records Joshua as being born/baptised on various dates - and I would only believe the register extract that says he was baptised on 9th January 1780. The other entries look as if they were made by amateur family historians who misread the month or year from the register and/or failed to distinguish between birth and baptism. A similar check shows that the Hemel Hempstead marriage register has been indexed for the relevant period and did not include the Joshua Ivory/Mary Puddephatt marriage. Whether the amateur entries have recorded the wrong parish or the systematic indexing of the register missed an entry is not clear - so the register microfilm must be checked.
Assuming that the marriage entry is correct and there are no earlier obviously relevant entries on familysearch of the British Vital Records Index, it would be best to see what other documents may exist which could give some clues to who your Joshua Ivory and Mary Puddehatt were, as even indirect information may give a clue as to where to look.
You don't give any indication of their age but it might be relevant that a Mary Ann Ivory, aged 85, was buried at St Mary's, Hemel Hempstead on 22 January 1842 [National Burial Index]. If it was your Mary she would have needed her parents consent to get married in 1773 - so there may have been an marriage licence. Have you looked at the microfilm of the register to see if they were married by licence - and if they were HALS may have a copy of it. However it may well not be your Mary as Ivory is not an uncommon name in the area, and your Joshua was not buried at St Mary's between 1800 and 1851 (although if he died before 1800 you would need to check the register microfilms.) While the information on the 1841 census is somewhat limited it could be worth checking the microfilm of 1841 census for Hemel Hempstead as Mary Ann Ivory would still have been alive, and may have been living with some of her children - which might prove whether or not she is your Mary.
The militia lists for Hemel Hempstead cover the period from about 1758 to 1786 and an index is available from the Hertfordshire Family and Population History Society. This will hopefully list Joshua and provide his age, and dates (at least when he was of an age to be in the militia) that he was in Hemel Hempstead.
In addition to making microfilms of the parish registers of most Hertfordshire parishes available through their family history centres the Latter Day Saints have also microfilmed many other records that may help build up the background to- if only by giving information about dates, social status and occupation. These are indexed under the library at familysearch. It would also be appropriate to see if there are any relevant wills at HALS or perhaps in London. There may also be some civic records (perhaps some at HALS and most with the Dacorum Borough Council) but they are probably further down the line if other things fail to produce results.
I see you are already aware or the Rootsweb IVORY-L list. In addition I see that a Sandra Carter (s.carter @t effect.net.au) has been posting to the soc.genealogy.britain newsgroup about her Ivory ancestors from Great Gaddesden and I wonder if you have been in touch with her.
Sorry I can't be more positive. I hope I am wrong - but I suspect you have got to the inevitable hard grind for slow (if any) progress in this ancestral branch.
There is a web page for Hemel Hempstead
If you can add to the information given above tell me.
Link updated January 2007