I would love to read a Timeline about a surviving al-Andalus, but I think it would also be rather difficult for them to avoid being constantly beset by crusades and diplomatically isolated from the rest of Europe. Probably control over the whole Iberian peninsula would be necessary, to keep the Pyrenees as a very defensible border and to stop continuous riding from Castile, Leon and the other small Christian kingdoms.
As for the consequences: they would totally change World history, but I think we cannot elaborate much on them without creating a scenario in which Islamic Iberia is viable and unified instead of being a collection of petty Emirates.
Possibly they would develop a variant of Islamic thought more influenced by classical philosophy? Ie Averroes has a bigger and more long lasting influence? What would their relationship with Morocco be?
It would be great if they created a culture that is both Islamic and European, and become a great power (Andalusian colonies in Americas are beyond cool obviously!).
I think that, if they survive, it means that that Islam and Christianity have a better reciprocal relationship, or religion plays a lesser role in European politics since WAY earlier, so there are much less crusedes and religion wars, but I am at loss to speculate how it could realistically come to pass...