Eileen, I'm also nowhere with those two boys, but have you found their elder brother Joseph who was born about 1847 at Slago Ireland?
I found who I think is him at Peckham, London in 1891 and 1901. He's a printer compositor age 44 and 54 respectively, with wife Alice born Salford Lancs age 39 and 49, son Benjamin born Hulme (nr Manchester) age 19 and 29, and son Richard born IOM age 10 and 20.
The FreeBMD shows a Marriage for Joseph Richard ATKINSON in the June 1870 quarter for Salford, which also lists an Alice Haslam BEARDMORE. There's a death for Joseph R ATKINSON age 73 Dec 1920 quarter in Camberwell RD, which includes Peckham. There's a death for Alice ATKINSON age 56 Dec 1907 quarter Camberwell RD.
I had no luck finding the family in 1871 or 1881. However, I would think the Richard ATKINSON age 2 listed at the bottom of grandparents Benjamin and Catherine Atkinson in 1881 at Douglas would be the son, though the age doesn't quite match 1891 and 1901.
After I wrote the above, I found on the IGI a Richard Atkinson born 15 Aug 1880 at Virginia, Braddan, son of Joseph Richard ATKINSON and Alice BEADMORE. This is where Joseph's brother John (the policeman and your ancestor) would have been, as his son was born there in 1879, though by 1881 they were in Everton. Richard Atkinson is baptized 3 Nov 1880 St Peter's Liverpool and it provides a matching birth of 15 Aug 1880.
Later again .... found family indexed "Allenson" at Osward, Chorlton cum Hardy, Lancs in 1871. Horrible handwriting. Joseph is 24 Printer born Ireland. Alice is 19 born Salford. There's also an Eliza [I think] something, looks like it could be "Beardmore" though indexed "Brandom" age 9 "Sister".
Have given up on 1881. They "ain't there".