To protect your band in United Kingdom you can register your trademark in two ways. Either through a direct registration in United Kingdom which can be as a direct filing or designating United Kingdomin an international registration. Or through a European Union trademark registration which grants rights in all 27 member countries of the EU. This is explained here).
More than one person can own a trade mark as long as it is used for a different type of product or service.
Fees associated with registration of IP rights in the United Kingdom are available in the wipo fee calculator.