I'm not a historian, however I do sing this specific music. I study this music, as well, so I have some knowledge about it.
Firstly, Jewish people were very involved in the culture of the countries they resided in, wherever they were. This obviously extended to the music that they listened to . Music is a very important part of the Jewish culture, as songs are sung on all occasions, including, but not limited to; prayers, weddings, brit milah, and any other group occasion.
Secondly, the Jewish people, as a religion, are a very traditional people. Rituals and traditions are passed down and are adhered to tenaciously. There are countless rituals that may seem strange to the modern viewer, however, the Orthodox Jew will carry on the tradition of his father, and his father's father before him.
These traditions will include actual rituals^1 , such as Kapparot, and Tashlikh, as well as, the way certain actions are performed, such as the tunes for prayers and hymns.
Moving on. Not to discount the lesser-known areas of Jewish musical culture, such as Yemen, Bukhara, and Ethiopia, we can put aside the European part, and the Middle Eastern part (i.e. Turkey, Iraq, Iran, Egypt, Syria, etc.) of the Jewish Diaspora, we can focus on North Africa (mainly, Morocco, Algeria, and Tunisia), where Andalusian music lives until today.
As we may know by now, Andalusian music is divided up in to the Nuba (Musical mode), as well as the rhythms. Traditional Moroccan Jews will use tunes from the traditional "Ala" in their prayers and hymns, which include the Baqashot. Some lyricists may also apply new words to non-jewish songs, to "hebraize" them for the general jewish population.
Finally, modern Israeli music has been heavily influenced by all Arab music and the Al-Andalus as well^2 , due to the influx of Arabo-Jewish immigration within the last century.
Tangent: There are a few songs of Al-Andalus origin that have actually spread to most Middle Eastern Jewish countries, a famous one is, Lama Bada Yatathana - - and the Hebraized version (ironically by a Morrocan Jewish cantor Haim Louk). There are others, of course..
Some more sources:
The Concise Garland Encyclopedia of World Music, Volume 2 page 794+
For a concise, yet clear analysis of the Al-Andalus influence on modern Jewish music you may check out http://www.jewish-music.huji.ac.il/content/andalusian-nuba
^1 some more customs of European descendant Jewry http://www.featureshoot.com/2015/04/a-fascinating-glimpse-at-the-rituals-of-strictly-orthodox-jews-living-in-israel/
^2 The Concise Garland Encyclopedia of World Music, Volume 2 pages 914-915