There is also a Lydia Shepherd/John Dyer marriage on the IGI in Portsmouth in 1791, and children born in London and christened up past John's marriage to Eliza Seager (but maybe born before and not christened until later?)
Hopefully, the original John Dyer/Eliza Seager marriage would say if he was a widower or not.
Children as far as I can find:
Lydia, 1796
John Jones, 1799
George Shepherd, 1801
Henry Anderson, 1802
Samuel, 1804
William, 1805
Ann Emes, 1808 (probably 'Ann Jones', see marriage below).
Emma, 1810
Now, Lydia through Henry were baptised at the Greenwich hospital, and the later children at the independent chapel.
There is a 1819 record in The Times for the marriage of a Lydia Dyer, daughter of John Dyer Esq of Royal Hospital, Greenwich, listed as 'third daughter'. (Married Lieut Samuel Meredith, RN). (different Lydia?)
The following are definitely this family:
The Times, Tuesday, Feb 06, 1838; pg. 7; Issue 16645; col F
On the 27th ult, at his residence, St. George square, Edinburgh, Dr. Henry Anderson Dyer, third son of John Dyer, Esq, Chicklade-lodge, Wilts, and late Chief Clerk of the Admiralty, in the 35th year of his age, universally beloved and lamented by his numerous family and friends.
The Times, Monday, Jul 29, 1839; pg. 7; Issue 17106; col D
At Greenwich, on the 27th inst, Mrs John Jones Dyer, of a daughter
The Times, Thursday, Sep 02, 1841; pg. 7; Issue 17765; col E
On the 30th ult, at Mere, Wilts, the lady of John Jones Dyer, Esq, of a son.
Found his death!
by looking for the Wiltshire address
via
http://books.google.co.ukGentleman's Magazine (1848)
1847 deaths, I think
Jan 6 - at Upper Clapton, aged 79, John Dyer, Esq, formerly of Chicklade Lodge, near Hinton, for many years Secretary of the Royal Hospital for Seamen at Greenwich, afterwards Chief Clerk of the Admiralty, and one of her Majesty's Justices of the Peace for Wiltshire.
There is also a mention of him being at Chicklade Lodge in 1825 on Google.
And in the Gentleman's Magazine of 1850 - marriage of Ann Jones Dyer, fourth daughter of the late John Dyer of Chicklade-lodge, Wilts, formerly chief clerk Adm and JP, to George Wailes, a solicitor of Newcastle-upon-Tyne.
Now, the mention of Lydia being third and Ann being fourth daughter suggests there were two earlier daughters, which might fit with Lydia Eliza Shepherd being the wife - and going by Eliza - as does the middle name of George.
http://www.nationalarchives.gov.uk/documentsonline/ has his will - John Dyer, 1847, Upper Clapton. That should give you some more info on his relatives, hopefully.
http://www.rootschat.com/forum/index.php/topic,284316.0.html - John Jones Dyer's census records over here.