No

The changes in the ionosphere caused by the total solar eclipse will  NOT affect either television or "FM" radio stations broadcasting . 

This means NONE of the national BBC stations nor the  regional and commercial radio stations  operating  at these frequencies will notice any difference  in reception during the eclipse. 
 

Because.....

This is because television and the radio stations that use Frequency Modulation (FM)  are are using much higher frequencies than those that are directly affected by the ionosphere.  Television  transmitters generally broadcast anywhere between 100 MHz and 1000 MHz and FM stations are broadcasting between 80-110MHz.

The radio waves at these frequencies shine straight through the ionosphere and tend not to be reflected like the lower frequencies .