In the UK, trade mark rights are estabalished in two ways.
One way of establishing trade mark rights is via use. If you use a trade mark (and marks include names, logos etc) on a reasonable scale so that you establish goodwill in the mark then you will have estbalished common law rights to the mark. Essentially goodwill in a business associated with a brand is established when a reasonable number of people know you by the that name. The following links may help:-
http://www.pinsentmasons.com/Default.aspx?page=1205 (click on the basics link - the first one)
http://en.wikipedia.org/wiki/Passing_off
The second way of establishing trade mark rights is via registration. You establish statutory protection to a mark via registering the mark at the relevant trade mark office, for the UK, the UKIPO, for all of the EU, OHIM and via other offices in other countries. Trade mark rights in the UK are not established by simply having a company registration. Trade Mark rights are not estabalished by registering the mark on certain commercial registers which are promoted on the net - they are only established via use and via registration at the relevant official IP Office.
Now turning to your issue. I presume you have traded under the mark on a reasonable scale prior to the other trader (sometimes 6 months may not be enough it depends on your level of use - prior passing off rights are not established by simply using the mark first, the mark must be used on a reasonable scale to establish goodwill). Also I presume the mark is distinctive enough to be associated with your company. However, if you have only traded in a local area, then the limitations of passing off mean you may have only estabalished prior common law rights in one locale. If the other trader has traded in a different part of the country where you were unknown, then the common law rights will co-exist. If this situation has occured in your case simply registering the mark first does not mean you will be able to stop the other trader using their brand. They will have the prior common law rights in one area which means you can not stop them legitimately doing what they can do under the tort of passing off and they could successfully oppose your application on the basis of those rights as mentioned by the UKIPO to you. They couldn't stop you using, as you would have the prior common law rights in another part of the country, but they could stopping you registering the mark as UK Trade Mark Registrations are national rights.
However, if your company used the mark on a national scale at the outset then you probably would have estbalished the prior common law rights. Also registering the mark first will also give you 'first claim' to the statutory rights. The otherside can oppose your application and opposition proceedings can takes years to sort and be costly. A link to a decision on a similar situation to your issue is below for illustration:-
http://www.ipo.gov.uk/o18907.pdf
However, if you had the prior common law rights you would win an opposition and succeed in registering your brand.
You could regardless of the registration situation try to stop them using now via a passing off action. However, its better to get a registration if you can as that makes it easier and cheaper to enforce your rights. To be successful in a passing off claim you have to establish the 'trinty of passing off':-
(a) you have goodwill in the business associated with the brand/name,
(b) there has been a misrepresentation - this really is simply confusion. You do not have prove fraud and ignorance is no defence,
(c) you have or are likely to have been caused damage by the othersides use of the conflicting name/mark. Usually if you prove (a) and (b), (c) is presumed.
Therefore the answer to your question is it depends and will very much depend on when, where and on what level each party has used their marks. This is a complex question and I strongly recommend you consult a trade mark attorney to get a detailed opinion. Links to some webistes of Trade Mark Attorneys for review:-
http://www.itma.org.uk/links#members-services
Hope that helps.