Following the convention, Basil is one of Ernest O’Bryen’s brothers.

I find these two (B&H) rather curious. There is an eleven year age gap between them, although the marriage seems to have been sanctioned by the family; and Harriet is a party to an Indenture of Settlement dated 13th May 1870, written at the same time as John Roche O’Bryen’s will. 

They marry almost exactly six months after JROB’s death, and therefore six months and two weeks after William Burke’s death. This would be the expected period of “full mourning” after the death of a parent, but still relatively soon after both fathers’ deaths. It might explain the somewhat low-key, and under-reported, wedding

Having said that, one or both of them are sufficiently well-connected for them to be married by the Archbishop of Westminster himself. The Pro-Cathedral was, what is now Our Lady of Victories on Kensington High Street.

from The Medical Press and Circular Advertiser Feb 8 1871

On the 1st inst at the Pro Cathedral Kensington, by his Grace Archbishop Manning, assisted by the Rev Fathers Foley and Conolly. Basil, second surviving son of the late John Roche O’Bryen Esq MD to Harriet Matilda, youngest daughter of the late William Henry Burke; both of Thistle Grove South Kensington.

Memorial to Harriet O’Bryen, Clifton Cathedral – 1876

from The Tablet Page 18, 5th February 1876

THE PRO-CATHEDRAL—A pleasing addition has lately been made to the Pro-Cathedral of Clifton. The side chapel, dedicated to St. Joseph, has been entirely renewed and decorated, and a marble altar erected, the reredos of which was executed in Belgium. The whole has a very pleasing effect. It is the gift of Basil O’Bryen, Esq, as a memorial of his late wife Harriet Matilda O’Bryen, who died August 23, 1873, and whose remains are buried in the cemetery at Fulham.

This is also curious. There doesn’t appear to be a huge family connection between the Burke family, and Bristol.  John Roche O’Bryen had practiced medicine there from at least 1841, and Basil himself was born there, but by the time he was 10 the O’Bryen family was in Liverpool, and by 1861, when he was 12, the family were in London, and he was at boarding school at Ratcliffe College.