A lot is owned by the Crown (which is shown on the map), which isn't exactly the Queen but more accurately the state. The royals do own a lot of land themselves too. I'm surprised the map doesn't show holdings from, for example, The Duchy of Cornwall, which is Prince Charles's estate, and includes massive holdings around the country (not just in Cornwall), such as most of Dartmoor.
Basically the answer is that the Duchy pretends that it is simply a private person (Charles Windsor) and so it shouldn't be subject to any more prying into its affairs than would be the case for you or me. This is sufficiently arguable that it doesn't fear ridicule.
A lot of that is public knowledge, but not on any data sets (I'd assume because most land registry records are still on paper). For example, near where I grew up a whole village is owned by aristocracy:
All of the Land Registry is computerised. If you can point at a precise piece of property you can go online, right now, to their site and pay them a few quid for a PDF of the data. If you need to show it to a court of law (e.g. to prove who owns something in court) you can pay a bit more and get a physical piece of paper issued which the court will accept as a substitute for a person from the Register showing up in person and telling them who legally owns it.
However, firstly (least importantly) not all land in England is registered. About 10-15% isn't registered. Registration is compulsory for new transfers today, but there are a large tracts of land whose owner hasn't changed for decades or even centuries, registration offers some benefits, but they're not obliged to register unless they want to transfer the property.
Secondly, the paperwork may just say a place is owned by a corporation in a tax haven. Knowing the land my building is on is owned by "Property Holding Corp Sierra Sixty Eighteen" in the British Virgin Islands is almost exactly the same as not knowing who owns it.
Royalty in the UK would presumably mean just the members of the royal family, which has a fuzzy boundary. The aristocracy is a much larger group, obviously.
Yeah, that's why I brought it up. For some 'royalty' might mean any sort of nobility while I took the original poster's meaning as the House of Windsor. I would expect that much of the land in England is held by various members of the wider aristocracy.
I make the distinction because as an American, many of my fellows conflate the two.