This question has the [united kingdom] tag, so the information about USA or other law and procedures is probably only of tangential use. Except for understanding that no, this is not something to ignore. It may well indicate someone trying to use your id fraudulently, or some other sort of data-processing foul-up that may adversely impact your credit rating.
The first thing I would do is phone the credit card company that sent the letter to inform them that I did not make his application, and ask firmly but politely to speak to their fraud team. I would hope that they would be helpful. It's in their interests as well as yours. (Added later) By the way, do not trust anything written on the letter. It may be a fake letter trying to lure or panic you into some other sort of scam, such as closing your "compromised" bank account and transferring the money in it to the "fraud team" for "safety". (Yes, it sounds stupid, but con-men are experts at what they do, and even finance industry professionals have fallen victim to such scams) So find a telephone number for that credit card company independently, for example Google, and then call that number. If it's the wrong department they'll be able to transfer you internally.
If the card company is unhelpful, you have certain legal rights that do not cost much if anything. This credit company is obliged to tell you as an absolute minimum, which credit reference agencies they used when deciding to decline "your" application. Yes, you did not make it, but it was in your name and affected your credit rating.
There are three main credit rating agencies, and whether or not the bank used them, I would spend the statutory £2 fee (if necessary) with each of them to obtain your statutory credit report, which basically is all data that they hold about you. They are obliged to correct anything which is inaccurate, and you have an absolute right to attach a note to your file explaining, for example, that you allege entries x,y, and z were fraudulently caused by an unknown third party trying to steal your ID. (They may be factually correct, e.g. "Credit search on ", so it's possible that you cannot have them removed, and it may not be in your interests to have them removed, but you certainly want them flagged as unauthorized).
If you think the fraudster may be known to you, you can also use the Data Protection Act on the company which write to you, requiring them to send you a copy of all data allegedly concerning yourself which it holds. AFAIR this costs £10. In particular you will require sight of the application and signature, if it was made on paper, and the IP address details, if it was made electronically, as well as all the data content and subsequent communications. You may recognise the handwriting, but even if not, you then have documentary evidence that it is not yours. As for the IP address, you can deduce the internet service provider and then use the Data Protection act on them. They may decline to give any details if the fraudster used his own credentials, in which case again you have documentary evidence that it was not you ... and something to give the police and bank fraud investigators if they get interested.
I suspect they won't be very interested, if all you uncover is fraudulent applications that were declined. However, you may uncover a successful fraud, i.e. a live card in your name being used by a criminal, or a store or phone credit agreement. In which case obviously get in touch with that company a.s.a.p. to get it shut down and to get the authorities involved in dealing with the crime.
In general, write down everything you are told, including phone contact names, and keep it. Confirm anything that you have agreed in writing, and keep copies of the letters you write and of course, the replies you receive.
You shouldn't need any lawyer. The UK credit law puts the onus very much on the credit card company to prove that you owe it money, and if a random stranger has stolen your id, it won't be able to do that. In fact, it's most unlikely that it will even try, unless you have a criminal record or a record of financial delinquency.
But it may be an awful lot of aggravation for years to come, if somebody has successfully stolen your ID. So even if the first lot of credit reference agency print-outs look "clean", check again in about six weeks time and yet again in maybe 3 months.
Finally there is a scheme that you can join if you have been a victim of ID theft. I've forgotten its name but you will probably be told about it. Baically, your credit reference files will be tagged at your request with a requirement for extra precautions to be taken. This should not affect your credit rating but might make obtaining credit more hassle (for example, requests for additional ID before your account is opened after the approval process).
Oh, and post a letter to yourself pdq. It's not unknown for fraudsters to persuade the Post Office to redirect all your mail to their address!
