I've dealt with this exact issue before, although through US law and not UK law. One thing I found is that a lot of places who have trademarks have to send out notices like this in order to maintain their ownership of the trademark.
That said, it's hard to get context on this with the information you've given us. Did this message come from a lawyer, or just an executive working there? A name change can kill a company, and the burden is likely on them to demonstrate that you are directly impacting them, which is hard to say without knowing anything about them. Do you think they have the resources to bring an infringement suit on you?
Trademark infringement can be hard to establish. A quick example: in the US, there are over 30 trademarks on the letters "AMD", for all kinds of different purposes. Trademarks are more than just a name, an infringement has to meet multiple standards.
I would look at the law on the books (in the US it's at uspto.gov, I don't know about the UK) and proceed from there.