There are plenty of stations on medium wave such as Smooth radio BBC Five Live.I used to get these but the radio is dead no noise but FM is fine. Strange
It does sound as though the radio is faulty, possibly in the AM tuner section. Medium wave and Long Wave are both amplitude modulated = AM as opposed to frequency modulation, FM.
Blaupunkt offer a retro looking DAB radio, the Bremen, SQR46DAB. But cheap it is not.