Richard,
Many thanks for your detailed information about Arthur's family, and also my apologies for not replying sooner (family matters, of the living kind!).
First to answer your last question. I'm related to Arthur through his older brother, James (1860-1922). My paternal grandmother was Gertrude Denny (1906-1968), who was a grand-daughter of James via his son James (1885-1933). My interest in the Denny family is centered mainly on those who stayed in Nottingham, though I'm also quite familiar with their family in Coventry.
I don't know if you have access to Find My Past or any of the other sites that have digitized newspapers, but I'm going to transcribe an article which was in the Nottingham Evening Post, dated 1 February, 1902:
"LONG SERVICE MEDAL
Among the recipients of the medal for long service and good conduct in the current army orders is Colour-Sergent Arthur Denny, Derbyshire Regiment. Colour Sergent Denny enlisted in the Derbyshire regiment in 1883, and has served continuously in that regiment from that date. He was promoted sergent in December, 1888, and transferred to the 3rd Battalion, Derbyshire Regiment, in 1893 for duty on the permanent staff, being promoted colour-sergent in 1895."
I also have another child for Arthur and Annie, other than those you mentioned, and her name was Sarah Ann(e). Her birth was registered in 1891 under the 45th regiment overseas births (the same method as her other siblings). I also have a death for Sarah Ann(e) in 1893 in the Shardlow registration district.
So how are you related to Arthur? All the Richards I have in my Denny tree are from the 17- and 1800s
Take care.
Denise