No direct equivalent of the map you linked to Tulyar, AFAIK.
Scottish Census data maps here:
http://www.scotlandscensus.gov.uk/ods-web/data-visualisations.html
Scottish Household Survey stuff here:
http://www.gov.scot/Topics/Statistics/16002
Edinburgh City results for 2014 here:
http://www.gov.scot/Resource/0048/00486844.pdf
On p.76 it says 43% of Edinburgh residents have no access to a car, up from 40% in 2013. Highest % for 15 years, and well above the Scottish average of 31%.
Rather infuriatingly, the city report only mentions access to cars, possession of a driving licence, and notes the correlation between relative poverty and lack of access to a car, and the link between gender and driving licenses (men more likely to have one). Can't find any mention of public transport; walking and cycling are only mentioned in the context of sport and exercise!
This rather demonstrates the mindset of those who set the parameters for what will be measured in statistics, does it not? Car = prosperity; cycling/walking = sport and exercise.
Sigh.