If you read something online (such as what I'm typing now) you've actually downloaded that content to your computer (or other device) anyway.
However the legislation relating to terrorism doesn't refer to 'downloading'; it only refers to 'possession' and to 'collection of information'. Sections 57 and 58 of the Terrorism Act 2000 apply:
https://www.legislation.gov.uk/ukpga/2000/11/part/VI/crossheading/terrorist-offences
The material downloaded in the case your link refers to was The Big Book of Mischief, which has always only existed in the darker places of the internet. However many prosecutions in such cases seem to refer to The Anarchist's Cookbook, which originally existed in a print version and was even promoted as a Fathers' Day gift by W H Smith (who also pushed sales of it at Christmas). I've even seen it on sale, fairly recently, in a charity shop. So it's hardly a secret underground publication!
The Protection of Children Act 1978 (as amended) [together with Section 160 of the Criminal Justice Act 1988] doesn't refer to 'downloading' either. It only refers to 'taking', 'making' or 'possessing' indecent images.
(There is also separate legislation, Section 62 of the Coroners and Justice Act 2009, which deals with possession of 'prohibited images' which are non-photographic in their natures, such as drawing and paintings. However that requires that images must be 'pornographic', rather than just 'indecent', for an offence to occur).
In such cases simply (intentionally) viewing an indecent image of a child on a website (without actually saving it) is an offence because a court has ruled it to constitute 'making' an image. [(R v Jayson (2002)].