Short answer: yes, the radio signal could reach that far.

Long answer: the British government and allied governments-in-exile in Britain collaborated to establish anti-Nazi, anti-occupation, propaganda radio stations that could be broadcast from London to occupied territories across the Channel and North Sea. In the Netherlands this was Radio Oranje - which was directly controlled by the exiled Dutch government in London and gave news and encouragement on the resistance, as well as news on Dutch colonies (Dutch territories in the Caribbean were untouched by the Germans, of course) and surviving merchant navy and military units serving in foreign armies (the Free Dutch Forces.)

But your question was specifically about Anne Frank and the BBC. The BBC broadcast to the Netherlands in Dutch, offering different programming from Radio Oranje. They also broadcast propaganda in German (remember Frank spoke German.) This was often aimed at ordinary German people, as well as those in occupied territories, and would poke holes in German news reports, satirise German leadership and the Nazi Party, and exaggerate British and Allied military power.