Why is the assumption made that Don't mention it is old-fashioned in BE? Panjandrum said nothing of the sort, only that it sounded old-fashioned to him.
One of the difficulties about making statements about whether a phrase is old-fashioned is that not many people have access to a corpus of spoken English, since such texts are often restricted to researchers.
In the publicly available BNC and COCA corpuses, at any rate, there are 47 and 92 examples respectively.
(If anyone knows of the existence of a corpus of spoken English that can be accesssed, please let me know!)
I'm not sure whether to call the expression old-fashioned or formal but in my experience (BE) the phrase "You're welcome" is much more current and even that is eclipsed by the frequency of 'No problem'.