I don't know enough to be able to identify the model, but a convertible Morris Minor 5142BY was integral to the BBC Play for Today, Nuts in May -
http://www.screenonline.org.uk/tv/id/548293/index.html

The entire play is on YouTube in 6 parts and the Minor is visible from the start, here -
http://www.youtube.com/watch?v=7CQYj_hDG7k&list=PL61647E6D04514896&index=106&feature=- plpp_video