British Passport holders get free of charge visitor/tourist visa upon arrival in Dubai or any other place in UAE. The visa would be stamped on the British passport as they go through the immigration in Dubai. Visitor/Tourist visa is valid for 30 days from the day of arrival in Dubai (UAE) provided the British passport meets certain conditions as noted below
British Citizens must be holding a Valid passport at the time of entry and should be valid for upto 6 months from the day of arrival and if they are transisting through Dubai, then it must be valid upto 3 months atleast. If you intend to overstay or extend your visa then you must be contacting the Directorate of Residency & Foreign affairs to know the options!