Huxley86, good evening.
Notice that this is your first post on the site? if no one else does welcome to the board, loads of info on here.
From the image you posted, the material on the ceiling does indeed appear to be Artex.
Artex could and did contain a low, to very, very low level of Asbestos the Asbestos incorporated was so called "White" Asbestos, this material is the lowest in terms of danger, brown and blue Asbestos are the real killers.
Up until about [2000] 1999 actually, Artex had, as above the possibility of containing Asbestos, a so called ACM [Asbestos Containing Material]
If you have concerns, it is possible to easily have the Artex tested there are several UK sites on the web that will test samples for you at a reasonable cost, you take a scrape sample send to the test house and wait a result.
If the material does contain Asbestos as proven by the test then you will have to contact your Local Authority to see what their regulations are about the disposal of any Asbestos material.
As an aside? have a look on this site, use the "Search" function [located at the top right of this page] to get a load more down to earth information about ACMs and disposal Etc.
Ken