I take your point about the cap badge, but why would someone living in Chester join the South Wales Borderers? Maybe because he was born in Wrexham, but usually men volunteered with their friends to be together in a local regiment. Checking the regimental records, the 4th (Service) Battalion of the Borderers was in Cirencester until June 29th 1915 when it moved to Woking - this doesn't tie in with the date on my first postcard. It did however embark from Avonmouth to Mudros, arriving on July 12th - this could tie in with the second postcard.
The other option is that the photo is actually of my grandfather's brother John Thomas Bradshaw who was born in Wrexham in 1896, but also living at Gough Court in Chester at the start of the war.