Thank you all for your contributions to this discussion.
The original photograph (attached) belonged to my wife’s grandmother Adeline (Dolly) Parker, whose single name was Burton. On the back is written “Dolly’s parents”, so I’m as certain as I can be that this is Jesse and Jennie Burton, who were married in September 1902. The female does bear a strong resemblance to Dolly. Also on the back is the photographer’s negative number, which dates it to some time in 1904. I have not seen the actual catalogue details (the photographer’s records are preserved in Eastbourne); maybe it lists the name/address of the subjects which would confirm their identity. But still, I’m happy to accept that this is the Burtons.
It is possible, of course, that Jesse is not wearing his own medals or, indeed, jacket. They may well be his and he has mislaid one of the clasps; or he may have purchased/borrowed someone else's for the photo. However, apart from the one missing clasp everything else appears to be correct. One possibility is that this is a retrospective wedding portrait taken two years after the event, when they could afford it.
Jesse was born in 1867 and appears on the 1871 census. His father, Joseph, died in 1877 and Jesse is absent from the next three census returns. We now know where he was in 1901, but the previous twenty-odd years is a mystery.