I speak from the perspective of Australian law - the general thrust should be the same but the nuances will be different.
There are several different ways that intellectual property (IP) is protected, I quote from Australian Business Law, CCH Australia Ltd (my emphasis):
Sales literature, software and promotional film about the product may be protected using copyright; its shape or pattern may be registered using a design; its invention may be patented; its name may be registered as a trade mark; and confidential information about it may be protected by action for breach of confidential information.
Copyright exists in "works" (ie original literary, dramatic , musical or artistic works) and "other than works" (ie sound recordings, cinematograph films, television and sound broadcasts and published editions of works). As you say, a single word or even a short phrase cannot be protected by copyright but an artistic work like a logo incorporating words can, however, in that case it is only the logo that is protected, not the words within it.
They have no copyright in their name
Single words and other things (eg Cadbury's trademark of a specific shade or purple) can be protected by a trade mark. However, unlike copyright which automatically springs into existence upon creation, a trade mark must be registered in the relevant jurisdiction (the UK in your case); a US or Australian trade mark, for example does not prevent anyone dealing with it in the UK. In addition, trade marks are limited to a particular field - if Microsoft is a registered trademark limited to the field of computing (and I'm not saying it is or isn't); if you were to apply the word to your range of feather beds there would be no trade mark protection.
Trade marks are publicly registered; look it up.
If they do not have a trade mark then they cannot take action against you.
Registering a trade mark is complex and can fall over for a number of reasons. First, it has to be capable of "distinguishing the product" - if the word(s) you are using are not made up and given that there are at least 2 products in the market similarly described (yours and theirs) then it is probably not trade markable.
My advice - do nothing unless and until you get something more serious than a letter. If you have breached their rights then they need to prove it; which means issuing you with a summons. If and when you get that (and you probably won't) you can a) roll-over or b) fight.
Worst case scenario is they can seek an injunction on you to stop - if they head this way you can just ... agree to stop before going to court. Theoretically they could also sue you for the damage you have caused their brand which is ... two-tenths of bugger-all.
That said, I am not a lawyer, not familiar with the jurisdiction and not appraised of your specific details; pay some real money to a real solicitor.