Another good source of information on sweet chestnut is English Nature's research report 627 "The ecological impact of sweet chestnut coppice silviculture on former ancient,broadleaved woodland sites in south-east England."
This is available on line or Natural England will send you one copy free:
http://naturalengland.communisis.com/naturalenglandshop/docs/R627%20part%201.pdf
It has, for example, this about reducing chestnut in Blean Woods:
"It was decided to reduce the area of chestnut on the reserve; a reduction in domination, not a removal. The objective was to convert pure monocultures into mixed species coppice, allow some reversion to high forest and provide glades etc. The methods available for removing or killing sweet chestnut are, grubbing out stumps, spraying regrowth and weed wiping. Mostly used the weed wiping approach. This was a phased approach attempting to kill off 15-20%
up to a maximum of 40%. Then, in four years time go back and cut the regrowth treating a proportion of the regrowth. This phased approach is softer on the landscape, it does not create a ‘desert’ of cleared/weed killed stumps. It is also possible to watch for natural
regeneration and treat stools where there will be most benefit. Areas can be reseeded with seeds collected on the reserve."
I think, Martin, if you approach the matter with a fairly light touch and undertake any chestnut removal gradually over a period of years, biodiversity in your wood is likely to benefit considerably.
A few standing dead trees, whether from ring barking or other causes, are always a wildlife asset especially for birds like woodpeckers, bats and various fungi. I was once shown some smallish ring-barked birches in the Wyre Forest NNR that were nesting sites for lesser spotted woodpeckers.