Fathers could be listed on Birth certificates even for illegitimate births. Both George's and Frank's births are easy enough to find after that it gets harder.
How about this as a candidate for George?
Household Role Gender Age Birthplace
Fanny Pine Head F 44 Stoke St Gregory, Somersetshire
George Pine Son M 8 London
Samuel Pine Boarder M 17 Athelney, Somersetshire
"England and Wales Census, 1901," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:X94M-Z1B : 29 August 2015), George Pine in household of Fanny Pine, Taunton St James Within, Somersetshire, England; from "1901 England, Scotland and Wales census," database and images, findmypast (http://www.findmypast.com : n.d.); citing Taunton subdistrict, PRO RG 13, The National Archives, Kew, Surrey
Edit: Oh I see from the comments you were already aware of this record. The absence of Frank suggests to me he probably died very young.
There is no such thing as purely British or Irish genes. Some genes are more prevalent in certain areas so are labeled "Irish" etc. It doesn't mean that an individual has recent Irish ancestry. It's just giving a general picture.