Complex sentence: In 1944, under increased danger from bombing, the Proms moved again, this time to the Bedford Corn Exchange (home of the BBC Symphony Orchestra since 1941), which hosted the concerts until the end of the War.
Split into simpler sentences: In 1944, however, increased danger to the Royal Albert Hall from bombing meant that the Proms moved again, this time to the Bedford Corn Exchange. This venue had been the home of the BBC Symphony Orchestra since 1941 and played host to the Proms until the end of the war.