Thank you all for the information and help! I realised a bit late that I should probably have listed all of my sources! Here is what I had; I have only added the reference for Henry in the 1851 census, which I didn't know about (only 1861).
Selina RowEngland and Wales Census, 1861 (RG09 3952/59 3)
England and Wales Census, 1871 (RG10 5263/54 6)
Re: the O'Briens, I was able to painstakingly confirm this as Selina's father John Roe's death certificate lists an O'Brien grandson as the informant.
Henry TysonEngland and Wales Census, 1851 (HO107 2437/508 12)
England and Wales Census, 1861 (RG09 3953/77 9)
England and Wales Census, 1871 (RG10 5263/54 6)
Death Certificate of Henry Tyson
The death certificate corresponds to the Bootle Workhouse Henry Tyson. I had always assumed the family was quite poor, but as I've said, the sheer number of Tysons (and Rowes) in Cumberland confuses matters quite a bit. One of the possible fathers for Henry was a known local thief. If Catherine Gertrude Elmes is the right Catherine, and did not exaggerate the "Esq.", I agree that we could probably cautiously exclude Workhouse Henry.
Henry Tyson (son)England, Births and Christenings, 1538-1975 (1866 in Gosforth)
England and Wales Census, 1871 (RG10 5263/54 6)
England and Wales Census, 1881 (RG11 5197/76

England and Wales Census, 1891 (RG12 4322/71 14)
Sorry if the birth record isn't cited in the best way, but I never quite figured out what Ancestry was actually representing with that collection.
Catherine TysonEngland and Wales Census, 1871 (RG10 5263/54 6)
An interesting possible connection I've noticed is that Catherine Elmes seems to bequeath something to members of a Bell family; Henry (son), if I have the right person in 1881 and 1891, was a boarder with a Bell family.
The other two children, Hannah and James, have their births and deaths in the GRO, but I'm not sure of the details. I do have James Tyson's birth certificate from the GRO:
6th June 1873, Church Lane Egremont | James | Boy | Henry Tyson | Selina Tyson, formerly Rowe | Farm labourer | + The mark of Selina Tyson, mother, Church Lane Egremont | 9th July 1873 | John Walter Registrar
I'm not sure his death certificate will provide much extra information, and haven't ordered a digital image of it (yet), but it's in the GRO for the next quarter of the same year. Hannah was born in 1868 and died in 1869.
As a post-script, if anyone is interested in chasing up Henry's father, I am mostly (tentatively) convinced by a marriage between a Henry Tyson and Elizabeth Benson in St James, Whitehaven, on 2 Sep 1815, but have not found the corresponding birth certificate.