Neither Andrew nor Jane appear in the 1841 census.
In 1851 they appear at 1 Cattle Market St. Douglas HO107/2525 folio 267 page 14 as Lynch, both 30 and both born in Ireland. Two children, Jane 9 and James 4 were born in Douglas. Andrew is a Labourer Agricul.
In 1861 they are at 9 Hanover St Douglas RG9/4416 folio 10 page17
Andrew Lynchey 38 Marine Stone Dealer b Ireland
Jane 38 b Ireland
Sarah 19 b Douglas
James 14 b Douglas
Mary 10 b Douglas
Thomas 8 b Douglas
Agnes 5 b Douglas
Jane 3 b Douglas
Andrew 7 months b Douglas
No census has any parents or in-laws so no clue as to who the are.
However, in 1851 there is a James Irving lodger widower 55 Banker b England at Kirby Braddan folio 424 page 5
In 1861 he has retired and is in Bradda on RG9/4412 folio 36 page 1.
In 1861 there is also a Lilly Irving described as an Aunt, unmarried 78 pauper b Monaghan? Ireland. She is at Back Strand Street Conchan, and, although it is not clear, she probably living with Henry and Alice McIlroy as he was born in Ireland and is the preceding entry. The reference is RG9/4418 folio 78 page 5.