Well obviously I couldn’t resist digging into this one!  And I’m glad I did because it is very exciting – a great story, and although not guaranteed to be true, it checks out in varying degrees in three different ways.
My first search turned up a little boy born on board HMS Romulus during the Battle of the Nile, and christened Owen Nile Reardon Smith, so that was no good.
However this webpage: 
http://www.bristolslostpubs.com/page167.html contains an extract of the 1861 census for the Saracen’s Head Hotel, Temple Gate, Bristol as follows: 
Quote:
Henry H. Giles, head M 62, hotel proprietor, born at the Battle of the Nile, English (1st August 1798)
Charlotte Giles, wife M 45, hotel proprietor, Cambridge
Biddy C. Giles, dau U 2, Somerset Bristol
 It also lists the following as proprietors: 
Quote:
1858 - 77. Henry Horatio Giles
1877 - 87. Charlotte Giles
 Is that the information your contact already has?
The BMD index has the following death for 1871:
Quote:
Name: Henry Horatio Giles 
Estimated Birth Year: abt 1798 
Year of Registration: 1871 
Quarter of Registration: Jan-Feb-Mar 
Age at Death: 73 
District: Bristol 
County: Avon, Gloucestershire 
Volume: 6a 
Of course I did worry that he was a pub landlord (or hotel proprietor), as “born at the Battle of the Nile” just sounds like a great story to tell your customers, and you can write what you like on a census form!
Fortunately four of the ships at the Battle of the Nile have an alphabetical index to their muster books (which is unusual), and it turns out there are two candidate fathers in those four ships, both named Henry 
Giles, one in the Alexander (
entry 488, Henry Giles, from Windsor, Ordinary Seaman, age on entry: 33), and one in the Bellerophon (
entry 1044, Henry Giles, from Carlisle, Coxswain, age on entry (1796): 24).
Without needing to look any further, it is the 
Bellerophon that checks out.  In the Gentleman’s Magazine  July-Dec 1860 on p.210 in obituaries, we have:
Quote:
At Portsea, aged 89, Nelly Giles. She was on board H.M.S. " Bellerophon," Capt. H. Darby, at the Nile, and in all subsequent engagements under Nelson, and was a most useful nurse to the sick and wounded. Three days after the battle of the Nile Nelly gave birth to a son. The Government awarded her a pension of £17 per annum for life.
The Bellerophon of course suffered the heaviest casualties in the British fleet, with 49 killed and 148 wounded.
The Malton Messenger for 21st April 1860 puts her death in April 1860:
Quote:
Last Week, Nelly GILES died at Portsea, being nearly 90 years of age. She was one of the few surviving witnesses of the battle of the Nile, having been on board Her Majesty's ship Bellerophon, Captain H. DARBY, and in all subsequent engagements under Nelson. Three days after the battle Nelly gave birth to a son.
The story also appears in ‘Nelson's Friendships’ by Hilda Gamlin, Vol 2:
Quote:
Another who merits a place in the record of sea-service is Nelly Giles, who was on the Bellerophon (Captain Darby) at the Battle of the Nile, and at all subsequent engagements under Nelson. Three days after the battle she gave birth to a son. She was a most useful nurse to the sick and wounded. In later life the Government granted her a pension of £17 a year as long as she lived. She ended her useful career in July 1857 at Portsea.
It also appears in ‘The College, the Market, and the Court’, by Caroline Wells Healey Dall, p.164:
Quote:
There recently died, at Portsea, in England, a woman, ninety years of age, named Nelly Giles. She was one of the few surviving witnesses of the battle of the Nile; having been on board His Majesty's ship " Bellerophon," in the command of Captain Darby, and in all subsequent engagements under Nelson. 
During the action of the Nile, she was surrounded by heaps of slain and wounded ; and she nursed the latter tenderly, undismayed by the horrors of the scene. Three days after the battle, she gave birth to a son. 
The government, in consideration of her great attention to the sick and wounded, and of the assistance she gave the surgeons, awarded her a gratuity of seventeen pounds a year for her life.
If Nellie 
Giles really did receive a government pension, there should be something to follow up there.
Having written the above, I just did one final search on the IGI records.  The result ties in with Henry and Nellie 
Giles, but throws a bit of doubt on the birth date – 3 days seems to have become 6 months:
Quote:
HENRY HORATIO GILES,  Male     
Birth:  14 FEB 1799    
Christening:  20 JUL 1800   Saint Thomas, Portsmouth, Hampshire, England 
Parents: 
  Father:  HENRY GILES  
  Mother:  ELEANOR     
Source Information: 
 Batch No.:  C062513,  Dates:  1770 – 1812,  Source Call No.:  0919726,   Type:  Film,  Printout Call No.:  6901162,   Type:  Film, Sheet: 00
The birth date of 14th February 1799 is while the Bellerophon is still in the Mediterranean, and the christening is three months after she arrived back in England in April 1800.
The IGI also shows 
Henry Horatio Giles marrying 
ANN JOYCE  on  04 MAY 1818  at Saint Marys, Portsea, Hampshire, England 
A son 
HENRY HORATIO GILES has a Christening:  26 MAR 1847   Great Wilbraham, Cambridge, England 
Parents:  Father:  
HENRY HORATIO GILES,  Mother:  
CHARLOTTE 
It looks like Charlotte is a second wife – the christening at Cambridge ties in with her birthplace of Cambridge in the 1861 census above.
A grandson (by another son) might be HENRY 
HORATIO GILES   Christened:  10 OCT 1875   at Temple, Bristol, Gloucester, England 
Parents:   Father:  CHARLES 
GILES,  Mother:  MARY ELIZABETH     
  
So in summary, it looks like we have the father, Henry Giles, from Carlisle, born about 1772, coxswain of HMS Bellerophon at the Nile, the mother, Eleanor (Nellie) Giles, born about 1771, died in Portsea, April 1860, and the son Henry Horatio Giles, born HMS Bellerophon either 4th August 1798 or 14th Feb 1799, died in Bristol, 1871.
The story may have improved with age, but it does seem likely that Nellie 
Giles was at least pregnant with Henry 
Horatio during the Battle of the Nile, and that he was born on board before the Bellerophon returned to England.  If he was born on 14th February 1799, it is worth noting that Nelson had shifted his flag to the Bellerophon at Palermo on 1st February, and then to back to the Vanguard on the 12th (although Nelson himself might have been otherwise engaged).