As it's a mixer and you say the hot water pressure is fine, then your problem is not in the nozzle at the end of the tap.
Turn the cold supply off and remove the head of the cold tap to allow inspection.
It may well be a washer problem.. they do sometimes break up or become dislodged.
DIY how to tutorial projects and guides - Did you know we have a DIY Projects section? Well, if no, then we certainly do! Within this area of our site have literally hundreds of how-to guides and tutorials that cover a huge range of home improvement tasks. Each page also comes with pictures and a video to make completing those jobs even easier!