There are lots of Gauntletts in Hampshire.
The original parish register baptism entry would give John's father's occupation in 1816. You could order the film from the LDS to view at one of their Family History Centres.
Or, since you are in London, you could go along to the Society of Genealogists library and for a day search fee, look at the microfiche of the registers there. But I warn you there are a lot of them
In the mean time, it does look likely that John who married Jane was baptised in 1816.
I can confirm the baptism of the other John Gauntlett in 1808, born in 1795, from a transcript. At that date there are no occupations included in the register.
William and Mary had the following children baptised in Gosport:
James 2 Aug 1799
William 29 Dec 1805
Ann 9 Dec 1807
Ann 2 Jun 1809
James 14 Feb 1812
These were baptised in Portsea:
John bn 23 Jan 1795 bapt 7 Oct 1808
Nancy 19 Sep 1802
William bn 26 Dec 1796 bapt 7 Oct 1808
And William 17 Feb 1793, at St Thomas Portsmouth
But this could be two different Williams
William Gauntlett married Mary Joliffe 29 August 1791 Portsea
William Gauntlett married Mary Stoye 2 March 1805 Portsea.
There is no indication that the second was a widower. And there is no burial of a Mary Gauntlett that would fit the widower scenario. So two couples.
A William Gauntlett, victualler of Portsea, left a will, proved in 1796, copy available at Winchester. This looking promising. He was buried in Portsea 8 Apr 1794.
John and Lucy Gauntlett had the following children:
Lucy 16 Nov 1813
John 28 Jun 1816
Ann 11 Dec 1818
Elizabeth 13 May 1821 buried 8 Jun 1823
William 21 Jan 1825
Eliza 19 Jan 1827
Mary Ann 12 Apr 1829
Thomas Henry 9 Dec 1831
Henry 12 Jul 1833
George 7 Aug 1835
Nell