The next step seems to be to find a death/ burial for John which would give his age.
If the son was born in late November and the mother was a widow by census date 2nd April, then the death must have been registered in, most likely, the first or possibly second quarters of 1871. There is nothing in the deaths overseas on Find my past, so it presumably happened in the UK or possibly Ireland as the 2nd battalion 6th foot regiment were based there in 1870-4. The first battalion was based in India1868- 1871 and was never associated with Jamaica as far as I can tell.
Here are the possibilities in England from the GRO. I did not look at spelling variants such as Yeats etc.
YATES, JOHN 38
GRO Reference: 1871 M Quarter in MANCHESTER Volume 08D Page 159
YATES, JOHN 57
GRO Reference: 1871 M Quarter in WETHERBY Volume 09A Page 96
YATES, JOHN 60
GRO Reference: 1871 M Quarter in FOLESHILL AND SOWE Volume 06D Page 305
YATES, JOHN 82
GRO Reference: 1871 M Quarter in BOLTON Volume 08C Page 247
Second quarter
YATES, JOHN 23
GRO Reference: 1871 J Quarter in OLDHAM Volume 08D Page 411
YATES, JOHN 39
GRO Reference: 1871 J Quarter in ROCHDALE Volume 08E Page 55
YATES, JOHN 71
GRO Reference: 1871 J Quarter in DERBY Volume 07B Page 237
YATES, JOHN 90
GRO Reference: 1871 J Quarter in BLACKBURN Volume 08E Page 349
You can probably eliminate the very old men.
The 6th foot later became the Royal Warwickshire regiment but in 1857 they formed a second battalion in Preston Lancashire, so John may have been recruited there. In 1864 the second battalion were in Jamaica which may explain the newspaper mention of being clerk to the Military Secretary.
Some history and links
http://web.archive.org/web/20071216173149/http://www.regiments.org/regiments/uk/inf/006RWF.htm