Manchester is at the centre of a huge urban metropolis and you'll probably have to travel some way to find a green haven. I had a look for you on Rightmove, land < £80k within 40 miles of Manchester, but was surprised how little there was that might have been of interest to you. You can set up searches on Rightmove and Primelocation to get email alerts on anything matching your interest, it does turn stuff up you'd otherwise miss, much of it from local agents selling at auctions. Wales is a likely hunting ground, there are woods regularly coming up there and the prices can be more reasonable than elsewhere. The downside is they tend mainly to be conifers on slopes (Wales has a few of those), small level deciduous woods with good access are difficult to find, you'll need patience and can expect intense competition when you find a good one, so if it feels like the right one don't dither. Have the finances in place so you can act quickly.
No doubt you're referring to our 'hosts', woodlands.co.uk and rivals woodlands4sale, they do cater well for the smaller amenity market. Have you looked at John Cleggs? They tend to have larger woods, even 'forests' of hundreds of acres, but also have smaller ones. This one might be of interest, if at the outer edge of your search patch? Their slow tendering system of inviting offers is rather stressful, a fixed price or auction is quicker and more 'open'.
http://www.johnclegg.co.uk/uk-&-ireland-property-search/property-detail/?id=832&filter_region=2/7/0&filter_price=&page=1Be aware that if you were to get anything in a National Park or AONB like The Peak District, Snowdonia or The Lakes, it comes with strict planning laws; you'll be unlikely to get consent for a 'shed'. Wooded areas outside AONB's and 'Parks have a presumption in favour of a 'toolstore and shelter for forestry purposes' using planning prior notification, which opens the possibility of a little hut on site as a weekend shelter or a fixed caravan.
What size and budget do you have in mind? Most of us seem to have around the 3 to 8 acres size, as you'll have seen depending on the nature of the ground and trees prices can vary between £5K up to £12K per acre, small one or two acre plots can be even higher.
Good luck, patience and happy hunting, its worth it.