It all depends on how you define wildlife!
The woodland edge, wide open rides, clearings and widely thinned areas all support what might be described as a more 'showey' flora and fauna. (Very small woods effectively are all woodland edge.) These species are of two broad groups; generalists i.e. species that can live in a wide variety of habitats and are not restricted to woodland, these species are usually very mobile enabling them move between sites and exploit new patches of habitat. The second group of these showey species are woodland specialists, but they're the sort of creatures that originally evolved to exploit the temporary sun filled habitat patches that arise when a mature canopy tree, or group of trees, dies and for a while far more sun reaches the woodland floor until a tree grows up and fills the gap in the canopy - returning the area to the original deep shade. These species also had to evolve to be mobile so they could find and exploit new habitat patches. Often the wildlife that we are most familiar with as woodland wildlife are these mobile creatures - we see them when they're moving. These are the sort of woodland creatures that had evolved characteristics that enabled them to flourish and massively increase in number once humans started managing woodland and creating many more sun lit areas within woodland than wood have been the case in the original wildwood. (Don't start quoting Vera at me - it was a good idea, stimulated debate but it has been put to bed IMHO!! :-) )Of course the ultimate example of woodland management that reduces shade levels and favours these showey mobile species is coppicing.
However there is another group of woodland specialists. These species are the inhabitants of deep, littles disturbed shadey woodland. These species evolved in the original wildwood where conditions were relatively constant and so typically didn't need to be particularly mobile - and didn't evolve behaviour patterns that led them to spend time searching for new suitable habitat patches. These greatures of deep woodland include many species of snail, beetle, micro moth etc. All these creatures are pretty cryptic and spend almost their whole life in a very small area. I would suggest that the majority of these creatures require deadwood of one kind or another.
Most large woods now found in the UK are either currently managed or have, in woodland terms, only recently been left to their own devices to grow wild (typically less than 60-70years since management ceased). These woods therefore contain very little of the the generalists and the woodland species that were once 'glade' specialists but that now flourish in human managed woods - which have vastly more sunlit patches than the original pre-human wildwood. Unfortunatley most of these woods with little human input in the UK have not yet been 'little or umanaged' for long enough to build up a great deal of the many kinds of dead wood vital for the specialist of deep large naturally shdey woodland.
You need deadwood both in various levels of shade, moisture and exposure - on the floor but also standing deadwood reaching to the canopy - and of many diameters.
It's this variety of deadwood that is usually the limiting factor in 'little or umananged woods' in the UK. My advice is always to create as much as you can. Fell trees and leave some on the floor - different diameters (as these have very different temperature/moisture patterns within them) but also kill some trees (ring bark) and leave them standing. Even if you think you have a lot of dead wood compared to other woods you visit try finding pictures of virgin temperate woodland and you will be amazed at the volume of dead wood.
Sorry I know I go on! But essentially managed woods, or small woods with lots of edge relative to their size have lots of sunlight and harbour certain showey, mobile species. The conservation of these is realtively easy as their mobility allows them to colonise new suitable patches relatively quickly.
The woodland specialist of deep relatively stable woods are much harder to conserve because the conditions they need take much longer than a single human lifetime to acheive - it takes a long time for a tree to grow to large diameter, die and begin to rot.
So most unmanged large woods in th UK have neither large numbers of the showey mobile sun loving creatures that rose to prominance in our woodlands thanks to coppicing (and that we think of as the 'main' woodland beasts but nor to they have large numbers of the beasties of the original wildwood that need large areas of very old growth with huge quantities of all shapes, sizes and kinds of deadwood.
I would suggest that in a small wood - pariculary if you have a lot of boundary with open habitats such as fields then there may be more value in promoting sun light down to the woodland floor. You will see the benefits very quickly (in woodland terms)- but still always ensuring you're developing some deadwood.
In a large wood the creation of all manner of deadwood is (in my oppinion)the most valuable thing you can do. However this is a much longer term game - and of course that is why provision of this sort of habitat is so much less popular and is rarely begun - and of course if begun is rarely allowed to run it's course - woodland ownership changes, perceived priorities change, woodland conservation fads come and go. YOu can undo decades of work towards true old growth in a week or so, letting the sunlight flood in and the new owner sees the 'benefits' in the next couple of years as the butterflies and flowers increase.
Of course both sorts of woodland are needed - it's just harder to have woodlands with a continuity of aim that lasts longer than a human lifetime, and letting the sun in is so easy, so quick and gives you beautiful tangible results in just a few years!
Any takers for the long haul of promoting old growth in their bit??