It is a great feeling when you can buy a small piece of England/Wales (or even Scotland, but be extra careful there or you may find Mrs Sturgeon has plans to take it from you), however unless you are very lucky, be prepared for lots of disappointments before you get there.
I've had that same yearning to find a piece of woodland/farmland for most of my life. Once I saved up what I thought was enough, the hardest thing was finding a patch of ground for sale - it took me 6 years. The small amount that does come up each year is mostly sold at auction, often as Woody2Shoes says, to a variety of people all prepared to pay way more than the "average" price of farmland. Small plots often go for 3 or 4 times the price of larger ones - even more if someone believes that one day they will get planning permission to build there dream home.
Before you go to any auction make sure you read and understand the Title Deeds to the property, it is surprising how many people still feel the need to restrict what you can do after they have sold. You need to have ready cash for most auctions, check the terms, the buyer typically needs to pay 10% on the day and the rest within 30 days.
Think twice before buying land with trees growing close to public roads, it could cost you a small fortune if they need professional work doing on them now or in the future.
One of the best places to look for land is in farming publications where (when there isn't a global pandemic) you can often see land being sold in multiple lots, there may well be an odd corner that would suit you. The "Farmers Guardian" is one of the best, though not sure how far south they cover.
The Smallwoods organisation that marcos gave you the link to is a great place to start. It doesn't cost much to join, I joined several years before I had any land. Once you are a member you will get access to all sorts of information about woodlands. They run low cost training events such as non-commercial chainsaw use and you will receive invites to woodland events throughout the country. If you go to them you get to see other peoples private woodland and there are often people there who know a lot about trees and who are only too willing to pass their knowledge on for free. One of them may even know of a patch of woodland about to come up for sale ... Good luck !