The short answer is, it depends.
Prior to the conquest it is thought that in the iberian peninsula the roman rite was widespread but it lived together with many heresies the most spreaded ones being the donatism between the hispano-romans and the arrianism between the goths. There were a few councils to try to solve the christianity issues in the Visigoth kingdom, the visigoths also converted from arrianism to roman rite. Gonzalez Ferrin (2006) proposed that one of the reason why the muslim empire had an easy time conquering the peninsula was because many of these heresy issues were not solved, and the local population prefered the "simple" islamic faith over the trinitary christianity. He also proposes that the had to be true for Syria Egypt and other eastern roman provinces. He also theorizes that many heresies appeared as a political opposition to the roman rule. The idea connects with Soto Chica (2020) propossal that the goths were the germanic population most romanized and they created a law and government as similar as they could to the roman empire.
For this time period we can consider that the Hispanians weren´t very connected to the rest of the Christians.
When the Ummayad conquest occured many of the population converted to Islam becoming Muladies (muslim Hipanians) but some kept their religion paying an additional tax the yizia. If an adult man was Christian or Jewish it was considered a dimmi (people from the book) and could preserve the religion paying this tax. This christians would be called Mozarabs, because they would keep their religion but would culturally arabize. In this stage many of the christians under muslim rule could have their own bishops their own governors. The relations between the islamic rulers and christians will be generally good to the point that Elipando de Toledo bishop of Toledo will invent adoptionism which tries to merge the christianity and islam saying that Christ is not a direct son of god but an adopted one. This will cause a lot of turmoil in the iberian peninsula specially in Asturias kingdom with Beato de Liebana thesis. The good relations come from the fact that generally mozarabs didn´t revolt but the arabs, berbers syrians and yemeni did. So good was the relation that a Bishop in Cordoba started a revolt where the would go to the mosque insult Islam and be killed, so many people did that the emir hold a christian council to call those executions provoked by the want to suicide and make them a sin. The bishops agreed.
As we can see the christians under the muslim rule on the early stages have conections outside of their borders but in general will be more interested in the inner politics of the conquistadors.
After this point everything went south. The constant civil wars and the wars against the norther christian rulers forced a constant raise in taxes against the christian populations which only had two options flee or convert. The people who fled joined the christian kingdoms and the people who stayed abandoned the practises of Toledo. The kingdom of Asturias broke their religious submision to Toledo -- which was the main religious place in Hispania-- in favour to Santiago de Compostela and the path of Santiago. Soon the christians under islamic banner converted to catholic or to islam, and after this point the connections with the rest of Europe will thicken as Asturias or their successor kingdoms connected with the rest of the Christendom.
Books
Soto Chica, J (2020): "Los Visigodos hijos de un dios furioso" Desperta ferro ediciones.
Gonzalez Ferrín, E (2006): "Historia General de al-Andalus, Europa entre Oriente y Occidente" Almuzara.
Pelaéz, A. (2018): "EL CALIFA AUSENTE. CUESTIONES DE AUTORIDAD EN AL-ANDALUS DURANTE EL SIGLO XI" La ergástula.
Manzano, E (2019): "La corte del califa" Crítica
ACIÉN ALMANSA, M. (1994) "Entre el Feudalismo y el Islam Umar Ibn Ḥafṣun en los historiadores, en las fuentes y en la historia" Universidad de Jaén servicio de publicaciones
Edit: I forgot one
Rodamilans, F. (2011): "Herejías en la península Ibérica hasta el siglo IX" Ab Initio 2