You can get parish records prior to 1837 for baptisms, marriages and burials.
Don't know if it helps but Parloc (Parish locator) has two Hadleigh's and no Hadley. The Hadleigh's are in Essex and Suffolk.
It may not help if the person reporting the death was not accurate. I had one death certificate say county Durham but the marriage certificate said County Down - two totally different countries. They may have misheard the details or just given the town where they first knew they lived.
There is a Halam, Halloughton, Harworth, Hawksworth, Hawton, Hayton, Headon, Hickling. Hockerton, Holme, Hoveringham, Hucknall in Nottingham.
You could try
http://www.nottsfhs.org.uk/ and see if they could do a search of the baptisms for you or there maybe someone on the Nottingham board who could do the lookup - when you post give them all the information you have i.e marriage, census's and death info as then they can extend what you know.
You probably already know this but the 1841 census entry for them in Essex says that John, Matilda (Mary?) and James were not of that county, Charles however was.
The 1851 says Nottingham B.L (?) for John and Mary - I don't know what the B. L. stands for or even if that is the correct reading of the letters but maybe someone else can help.
I'd try Nottingham for the marriage and for a baptism for James.
Jenny