I know that downloading copyrighted content is illegal, and that the actual process of transmitting the copyrighted material is also illegal, but I can't find anything about actually viewing.
If this is illegal in the UK, what piece of legislation would specifically cover this?
Of course it is illegal, in the same way as using pirated software is illegal.
Standard copyright legislation applies.
And your statement about "download vs view" is utter nonsense.... do you think if someone "just views" kiddie porn they should get away with it, and only people who save the images to their computer should be locked up ? No ? Right, well the same applies to you and your bypassing of distribution mechanisms to view of TV programmes.