Just because your using the same generic term does not mean they have anymore right to the .co.uk than you…? there is nothing stopping you from filing against them for the generic .com
The law permits businesses in different geographic markets to use identical trade names, unless the good will and reputation of an existing business extend into the market where a new business has opened
A word/term is generic if its primary meaning to the prospective purchasers is the product/service and not the producer of the product/provider of the service.
Words/Terms that originally are trademarks/marks may, over time, be adopted by the public as the common name for the product/service and become generic. If a word/term becomes generic, the owner of the trademark/mark cannot exclude others from using it or recover damages for the use. Examples of words that have lost trademark protection because they became generic names of products they identified include "aspirin", "thermos" or "cellophane."
These may help you figure out how generic the word is,,,, and if you case is good or bad in the uk
Intellectual Property Office - Trade marks are signs which distinguish your goods/ services from your competitors Intellectual Property Office - A domain name is a name by which a company or organization is known on the Internet Intellectual Property Office - Passing off unregistered trade marks