I've only seen greater spotteds in urban parklands (Hyde Park, Greenwich Park, Oxleas Wood) but their presence in urban areas may be more to do with a scarcity of food in an agroindustrial countryside and the (over)use of pesticides.
I come from northern 'hill country', so very much not agro-industrial, and not the sort of pesticide usage as one would get in commercial orchard and other horticultural areas; there have always been lots of woodpeckers in deciduous valley woodlands. As a child, my parents were keen bird-watchers and our home had a large garden dropping down towards some semi-ancient woodland which was FULL of woodpeckers. They could make a right racket all hammering together and you'd wonder how
they could not get a headache while being so good at giving
you one ... but we very rarely saw them and they never visited the garden although a very wide variety of other birds did. Finches of many different sorts, treecreepers, nuthatches, wrens, blackbirds, song and misselthrushes, redstarts ... but never any woodpeckers.
I do confess that in the few times I've lived 'down south' I'd always been surprised at the lack of 'wild' wildlife in the so-called rural areas - and gratified and delighted at its abundance in parks and gardens ...
In the relatively-pesticide free life of the uplands - which includes wooded valley portions of the uplands - I wonder if the apparent reduction/change in habitats in some bird life, at least, doesn't have a great deal to do with loss of mature trees
in the countryside over the past 40 years or so - the 'Great Storm' of 1987, Dutch Elm disease, Ash dieback, and the necessary clearance and tidying up deemed necessary after/during such events. Older, decorative trees in gardens may well have been the woodpecker's sole remaining source of food so the bolder ones ventured or were driven by hunger into the villages and outer suburbs and over the years they have thrived and spread more and more widely into well-treed urban areas. I imagine this may well have applied to other parts of England, too, with added effect in agro-industrial areas as the birds were 'forced' out of any small bits of woodland remaining, into the trees of urban and suburban parks and gardens, with concomitant lower exposure to the side effects of agrochemicals.
One of the other large factors, at least in upland areas, which is often not acknowledged for various 'sensitive' and even so-called PC reasons, in loss/disturbance of habitat has - I believe - been the increasing number of the public 'using' the countryside, especially when there is a sudden influx, as there often is, during periods which are sensitive to the wildlife in question. People can get really
really angry when asked to 'keep to the path', 'keep your dog on a lead', 'don't play in the brook', 'keep out of the water' etc etc - not realising, and/or not caring that ground nesting birds, breeding newts, fish fry, mating hares etc etc can
and are being damaged and/or distressed by such actions. The very large groups that can sometimes now be seen in traditionally-popular hiking areas are an additional pressure, as they spread out either side of the accepted, long-used walking/riding tracks - understandably so from their POV - but troublingly so for the generations of 'whatever' flora or fauna which have until very recently thrived within literally just a few metres of a well-trodden path.
I can perfectly well see how and why agro-industry can result in destruction of the environment needed by many species; however agro-industry is not the only factor resulting in changes and challenges to the rural environment.
Anyway this is very off-topic so I will stop now.