is it an offence specifically under the terrorism legislation to watch a video?
Viewing a video is not, by itself, a criminal offense.
Viewing a video that was produced to promote terrorism is not, by itself, a criminal offense.
Viewing a terrorist training video with the intent to commit terrorist acts is a criminal offence.
According to legal commentator David Allan Green, the relevant act is the Terrorism Act 2006 sections 1 and 2
The Terrorism Act 2006 has three parts, the contents of parts 1 and 2 are
Part 1 Offences
Encouragement etc. of terrorism
1.Encouragement of terrorism
2.Dissemination of terrorist publications
3.Application of ss. 1 and 2 to internet activity etc.
4.Giving of notices under s. 3
Preparation of terrorist acts and terrorist training
5.Preparation of terrorist acts
6.Training for terrorism
7.Powers of forfeiture in respect of offences under s. 6
8.Attendance at a place used for terrorist training
Offences involving radioactive devices and materials and nuclear facilities and sites
9.Making and possession of devices or materials
10.Misuse of devices or material and misuse and damage of facilities
11.Terrorist threats relating to devices, materials or facilities
12.Trespassing etc. on nuclear sites
Increases of penalties
13.Maximum penalty for possessing for terrorist purposes
14.Maximum penalty for certain offences relating to nuclear material
15.Maximum penalty for contravening notice relating to encrypted information
Incidental provisions about offences
16.Preparatory hearings in terrorism cases
17.Commission of offences abroad
18.Liability of company directors etc.
19.Consents to prosecutions
Interpretation of Part 1
20.Interpretation of Part 1
Part 2 Miscellaneous provisions
Proscription of terrorist organisations
21.Grounds of proscription
22.Name changes by proscribed organisations
Detention of terrorist suspects
23.Extension of period of detention of terrorist suspects
24.Grounds for extending detention
25.Expiry or renewal of extended maximum detention period
26.All premises warrants: England and Wales and Northern Ireland
27.All premises warrants: Scotland
28.Search, seizure and forfeiture of terrorist publications
29.Power to search vehicles under Schedule 7 to the Terrorism Act 2000
30.Extension to internal waters of authorisations to stop and search
Other investigatory powers
31.Amendment of the Intelligence Services Act 1994
33.Disclosure notices for the purposes of terrorist investigations
Definition of terrorism etc.
34.Amendment of the definition of “terrorism” etc.
35.Applications for extended detention of seized cash
Section 6 Training is
(2)A person commits an offence if—
(a)he receives instruction or training in any of the skills mentioned in
subsection (3); and
(b)at the time of the instruction or training, he intends to use the skills
in which he is being instructed or trained—
(i)for or in connection with the commission or preparation of acts of
terrorism or Convention offences; or
(ii)for assisting the commission or preparation by others of such acts
(3)The skills are—
(a)the making, handling or use of a noxious substance, or of substances of
a description of such substances;
(b)the use of any method or technique for doing anything else that is capable
of being done for the purposes of terrorism, in connection with the commission
or preparation of an act of terrorism or Convention offence or in connection
with assisting the commission or preparation by another of such an act or
(c)the design or adaptation for the purposes of terrorism, or in connection
with the commission or preparation of an act of terrorism or Convention
offence, of any method or technique for doing anything.
It seems questionable that a judge and jury are likely to be persuaded that viewing any video of a beheading constitutes receiving training and unless there is evidence of intent to use this training, no conviction could result. There are other sections but I see no explicit suggestion in the text that viewing a video of a beheading is itself a terrorist offence under this act. Obviously, interpretation of law is a matter for courts.
According to The Register
Watching the video could be used to build a case against us, we were told, but this act alone did not give plods enough evidence to lock the viewer in Belmarsh.
"Distribution is the issue," the spokesman said. "Viewing the video could be taken into consideration if any other information comes to light."
So merely viewing the video is unlikely to lead to arrest. Unless you have already committed other acts that can be construed as evidence of disseminating materials that promote terrorism.