Looked up newspapers and there were two short death notices in the Irish Times and Irish Independent both of which read the same
Tues March 3rd 1953
BRENNAN, March 2 1953 at his residence, Annacloon, Oughterard, Surgeon Captain, J.R. Brennan RN (Ret'd) Beloved husband of Rovena. Funeral arrangements later. English Papers please copy.
Wed March 4th 1953
BRENNAN, March 2 1953 at his residence, Annacloon, Oughterard, Surgeon Captain, J.R. Brennan RN (Ret'd) Beloved husband of Rovena. Funeral in Oughterard after service in Oughterard Church at 2.30pm today (Wednesday) English Papers please copy.
Followed by an acknowledgment on March 12th 1953
There did not seem to be any obituary or other newspaper piece at the time or any other reference to Joseph Roland Brennan at all.
I found a piece from 1918 in the West Cornwall News that I believe refers to Clarice as the soloist at a fundraiser in Praze.
The link that Heywood posted is probably the best avenue to follow to find out where in Oughterard Joseph was buried. His will was published in 1953 with Clarice R., his widow, the beneficiary of his estate worth £2,786/14/4 which was a substancial sum at the time.
The fact that Clarice was still quite young (55 approx) if she remained in Ireland her death maybe after 1959; this is the cut off point for searching on-line. You can search the original records by year which means either a long trawl or requiring further information.
There are
Wills on-line up to 1982 that can be useful to try and locate a person's death between 1960 and 1982, but only reliable for those that had effects that went through probate. Clarice does not seem to appear on this list.
Some options are that Clarice remarried or that she returned to England, however I would expect to see her death on the UK records by 2006 (aged 108). She may have imigrated to another country either!
I am not based in Galway so cannot look in the cemetery, finding someone who can to confirm Joseph's burial and who else may be in the same plot is probably where to start.