You're up against an NI company that operates in the UK and ireland.
I think it all comes down to what "X" stands for.
If it stands for "travel" I don't think they have a leg to stand on, regardless of when they set up their company, because it is such a generic term and it describes your business. Similarly if it stands for DunnesTravel and your name is Dunne and you sell holidays, you should be fine.
If it stands for "EdsList" and they had an established and reputable business called "Ed's List" that you should have been aware of, you're on dodgy territory.
I think you know yourself whether they have a case. Registering a business name doesn't mean anything. Even if you registered a company name you may still fall foul of infringing someone's intellectual property.