Hadrian’s Wall has no single owner, not even the UK government or its agency, English Heritage as was (now Historic England). Rather it belongs to the many landowners whose property it crosses. This of course includes the National Trust, who own much of the central sector after the Clayton estate was sold off in the 1920s. However, the largest single owner of the curtain wall at least might be Northumberland County Council, who own the Military Road (now the B6318) underneath which much of Wall Miles 12 to 33 are located (although how much survives is unknown).
Further reading: Symonds and Mason 2009