There seems to be only one possible candidate in Suffolk - bear in mind that surnames can vary between Mathews/Matthews, it's a common surname and also that the vast majority of baptisms would show just one Christian name and not 2. Indeed, we have found from posts on this site that people who had one Christian name in the UK suddenly have 2 when they emigrate! Perhaps because of something to hide or for some added prestige?? Who knows!
There was only one couple named William and Sarah in Suffolk, in the relevant time frame, having children baptised.
1841 Census - New Street, Woodbridge.
William Mathews 47 Publican Not born Suffolk
Sarah Mathews 43 ditto
Jane Mathews 13
Robert Mathews 11
Thomas Mathews 9
Mary Ann Mathews 6
Caroline Mathews 4
William Mathews 15
Babe not named (female) 3 weeks
(all the children born in Suffolk)
Can only find some of the baptisms:
William Matthews bp.12/6/1825 Shottisham, Suffolk
Frances Matthews bp.26/9/1827 ditto
Jane Matthews bp.11/10/1829 ditto
Robert bp.5/6/1831 ditto
Thomas bc.1832
Mary Ann bc.1835
Caroline bc.1837
Matilda Margaret bp.28/9/1841 Woodbridge (died Dec.qtr.1841 Woodbridge)
By 1851 William and 2 of his brothers are in Harwich, Essex:
William 26 Hair Dresser born Debenham, Suffolk (this is incorrect)
Eliza 30 wife born Woodbridge, Suffolk
Robert brother 19 Shipwright ditto
Thomas brother 17 Shipwright ditto
Cannot locate an obvious marriage for William and Eliza!
1861 - Harwich, Essex
William 36 Hair Dresser and Tobacconist born Shottisham, Suffolk
Mary Ann 35 born Woodbridge, Suffolk
Think this second wife possibly Mary Ann Kell married Tendring district, Essex in Dec.1857 (a Mary Ann Kell was born in Woodbridge).
However, by 1871 William has vanished!
1871 - 23 Church Street, Harwich, Essex
Mary Matthews Marr. 35?? (error) Hairdresser born Woodbridge, Suffolk
William John Fisk Servant 23 Hairdressers Apprentice ditto
Can find absolutely no sign of William Mathews in 1871 although Mary still shown as married.
By 1881 - still in Harwich - Mary Ann Matthews 50 now a widow.
There is no likely death entry for a William Mathews/Matthews between 1871 and 1881 (in Essex or Suffolk).
I suspect that between 1861 and 1871 William left Mary Ann and made his way to Australia to make a new life. Although having had 2 wives at this point he had no children with either. After a wife had been left for 7 years she could legally have her husband declared dead which is what I suspect happened here.
I feel sure that this William is your man.
As to his parents, William and Sarah. The 1841 Census (which didn't state actual birthplaces) simply states that although living in Suffolk neither of them were born in Suffolk.
By 1851 sons William, Robert and Thomas were in Harwich, Essex but William and Sarah plus their daughters are still living in Woodbridge plus they have a further son Henry bc.1843. William is shown as bc.1793 Crondall, Hampshire and wife Sarah bc.1798 Godalming, Surrey.
Because of the common name, and with the 2 of them coming from different places, they could have married anywhere. There is a marriage of a William Matthews bachelor to a Sarah Peache, widow, 1/12/1822 Godalming, Surrey (Sarah's birthplace) which could possibly be them, making her a 24 years old widow.
The birth certificate of one of their 2 youngest children would confirm Sarah's actual maiden name.
Hope this helps.
Annette