Nearly forgotten I wanted to contact the Archives about this but I have done so now. Hampshire Archives had nothing relevant and referred me to Portsmouth History Centre.
I today received an interesting reply from them - in the negative as far as burial records are concerned, but useful background. So with grateful thanks to Anna Delaney of Portsmouth History Centre, I shall quote or summarise her response.
Firstly PHC don't have any burial records for the military cemetery at Gatcombe, Hilsea. Further: "The National Archives only appears to have baptisms and marriages for the Hilsea Garrison from 1877 and the Church of St Barbara wasn't established until 1888."
The 1850 document by Henry Slight referenced in the linked Mailing List of 2002, is to be found at Portsmouth History Centre and is apparently hand-written with a challenging style! But from what Anna could make out, she transcribed:
"The cemetery of Gatcombe is at the very verge of the Municipal Borough at Hilsea. This had been a burial place for many ages… During the last century and part of the present it served a [like?] purpose for the [illegible] bodies of troops both [illegible] and militia who occupied the Barracks adjoining…After the peace of 1814 it was of an [illegible] and now in 1849 presents to view a large green meadow very much elevated above the turnpike and the adjoining fields. The field is now the property of the [illegible] Board of Ordnance. The funeral rites were performed by the military chaplains and no fees were paid."
This document has no details of individual burials.
End of their response
Again, grateful thanks to Portsmouth History Centre for that. At least we have confirmation that military chaplains were involved. Whether any of the various lists at Kew for Hilsea Depot might contain an elusive burial list in the middle of something else, I have no idea. I do know that WO12/12957 Hilsea etc Muster Lists, had, at the start of the "Q4" list for 1814, a loose sheet with a list of deaths at Porchester General Hospital for 1814 in regimental order. My guy wasn't in that lot, I'm just using it as an illustration that not everything in the box was what it said on the tin (to mix metaphors).