As deggers says a fridge should not blow a fuse. If it is 5 months old you have your statutory rights but Comet are not saying they will not send an engineer, just that it will be over a week. I think your best (quickest) bet, after the fuse has been checked is to use your negotiating powers with Comet and get them to send an engineer with some urgency, talk about a possible claim for damages and/or their poor reputation. Check your guarantee to see if they give a maximum time for an engineer to call.
Your payment by credit card will offer some protection as they can be held responsible for defective goods and services, if the cost was between £100 and £30,000 but a claim will take time and Comet are not refusing to provide a service, they are just providing a slow service.