They also seem to say "020" cannot be used as they think this will cause with "1800" and that the general public are likely to be confused.
They can demand that you stop using your business name due to "passing off", which as I have pointed out already is a concept in common law (for unregistered trademarks) which means your company sounds very similar to their established company and likely to cause problems for them with confusion among the public and possibly reduced trade.
Passing off causes a problem if you both operate in a similar or the same industry and if their established business has been trading for several years. The court also asks whether the name is "likely to cause confusion" among the general public. Geography comes in to it too - if the businesses are not miles apart then it may well cause an issue too. The court weighs up all these factors and if the established company sues, the court may well agree and tell you to change your company name.
There are no specific laws in relation to this - the passing off question has been devised by the courts over the years but bear in mind if they take issue with your proposed name then they will be able to ask the court for an injunction (which costs £308 to apply including their solicitor's costs and they will ask for an order that you pay, assuming their injunction is successful).
In terms of next steps you can propose to them that the word "disco" is a generic term and their client cannot possibly hope to allege you have infringed that element of the name. The "020" number is ***** different to 1-800 or 1800 as a prefix, but it would come down to the court's interpretation as to whether this is too similar and likely to cause confusion. It is an objective question - so the judge will ask themselves whether the 020-disco is too similar to 1800-disco. There is a chance that they would think it was one and the same company.
His company has only been running for just over 2 years - so it is not a long-established company. That will not help them. You can ask them why their client has failed to trademark to ensure his rights are preserved - I presume it is because he has not bothered or not appreciated the value of intellectual property in a name. Your defence would be that the claimant's mark is too generic with the word "disco" and the fact he is not running a well-established business, so proving actual loss will be difficult for them (and proving misrepresentation will also be difficult to do).
The best way forwards would be to point out the above in reply to the solicitors (this is only guidance by the way, not legal advice or intended to be taken as legal advice on this website) - and suggest that your client has not proven his loss (or a likely indication of his loss) among the other arguments I have listed. You can propose to change the numbers in the name, but that the word Disco will not be changed as it is a generic term which cannot be trademarked.
I hope this helps - if you can please accept the answer and give me a 5 star rating (there should be a button at the top of your screen to do this), I can answer follow up Q&A's at no extra charge and Just Answer will credit me for helping you today.