The best way for you to continue is to buy a copy of Ernest birth certificate. There is no Jack or Kathryn registered to a Green/Phipps partnership.
What is known is that Ernest was born in Ashton under Lyne on the 19 March 1929 and passed away in 2006.
https://www.gro.gov.uk/gro/content/certificates/April-June quarter 1929, Ashton under Lyne, Lancashire. Volume 8D page 1088.
As it was just after the Great War, Ernest mum could have had children by another man who was killed and so Jack and Kathryn would not be Greens.