Ubiquitous writer's beginning to mosey about in Edinburgh (8)
I believe the answer is:
immanent
'ubiquitous' is the definition.
'writer's beginning to mosey about in edinburgh' is the wordplay.
'writer's' becomes 'im' ('writer's' could mean 'the crossword setter is').
'beginning to' indicates taking the first letters.
'about in edinburgh' becomes 'anent' (Scottish dialect term for 'about').
The initial letter of 'mosey' is 'm'.
'im'+'m'+'anent'='IMMANENT'
(Other definitions for immanent that I've seen before include "everywhere in godlike fashion" , "(Of God) permanently pervading universe" , "Existing or operating within" .)