I think the order in which they put them is nonsense. If judging by most popular or beloved (or as you like), how can Saariaho even break the top 50, not even mentioning she was put ahead of Gershwin. Including John Cage (can anyone actually name anything other than 4'33 and maybe his prepared piano concept) Morton Feldman, and the most surprising Charles Ives (who I love dearly with all my heart), but hardly anyone even knows who he is.
If you really wanted to measure a composer by how "great" they are, their accomplishments and impact to the history of music, and legacy of their field should be the starting point. Therefore making (really hard to say in what order, probably not possible given a historical context)- John Williams, Pierre Boulez, Olivier Messiaen, Claude Debussy, L.V. Beethoven, Arnold Schoenberg, Karlheinz Stockhausen, Philip Glass, J.S. Bach (although a genius beyond measure, his music was hardly known at all until Mendelssohn brought him back, and even in his time he was considered kind of a weirdo at times- so not so sure he can be considered either). * Mozart and Haydn have been left of the list because their legacy was -seemingly- shattered by Beethoven- whom everyone copied for a great long time, however, all Beethoven really did was super stretch the boundaries of the "classical" period music, similar to Stravinsky in his time.
Now there are certain composers that were influential to some but not all : like Tchaikovsky to Stravinsky ( the Firebird really at moments sounds quite like the Nutcracker, which, in the entire Nutcracker ballet, the "non-famous" parts, often sound very very much, and even at times just taken directly, from Beethoven's 7th Symphony), and not sure I could say, but perhaps his big sweeping melodies influenced the Golden Age film composers. But this is not as easily documented.
Perhaps most people would not agree, but again its based on contribution. Olivier Messiaen influenced countless composers in his classroom, just as a very start. Stockhausen basically created "noise music", and furthered the boundaries of what could be done with electronic music (which can be seen still today which of course has genre crossed for the technique, not the music style itself), Schoenberg/ Boulez (although unintentionally for Boulez) created armies of disciples who continued writing in 12 tone/ serial fashion- Boulez taking it further with also his, in a different way, contributions to the electronic music field (with the hard work of the great fellows at IRCAM), Philip Glass also spurred a movement of composers, who didn't exactly steal his style directly, but the principle and harmonic progressions for sure (listen to the slow movement of his Tirol concerto, and don't tell me Hans Zimmer stole that and used it over and over again), Debussy (need I say it? how many times have you heard a composer say they were inspired by him- probably most of the people on that list, and then how many poor imitations or rip-offs of his music?).
And last of course, John Williams, who need no explanation, who I think in 100 years or less, be the hands down winner of the most influential of all time.
So finally, I think the BBC list just seems very random to me, without any kind of purpose at all. If people are going to make a list based on glib thoughts, and ignorance of a composers hard work and contributions (or there lack of) to their craft, then they shouldn't be making a list at all.......take THAT people who voted for Brexit!
(PS- The Sherman Brothers and Alan Menkan were not even listen, and they had a huge impact on the musical scene too, not to mention Andrew Lloyd Webber)