Wragg was brought up in Slough, England and started working in a bank as a young man. His hobby was model-making and when his father's company had a contract with Gerry Anderson's puppet film company AP Films, Wragg worked for them as an assistant and later as an effects designer on shows including Thunderbirds and Captain Scarlet.
Moving to the BBC, Wragg worked on many iconic programmes, such as Doctor Who, Allo Allo!, Filthy Rich & Catflap, Bottom and The Two Ronnies. He was recruited for Red Dwarf because of his extensive work with models.
Wragg created many of the show's models, including Starbug, Blue Midget and Red Dwarf itself. He was also responsible for destroying the Red Dwarf model for "Demons and Angels". One of his other creations was Kryten's make-up, for which he worked with make-up designer Bethan Jones. He enjoyed working with water, creating many underwater scenes and model shots. Craig Charles joked that Wragg needed little excuse for putting on his wetsuit, even doing it when it was simply raining.