I don't think there would be much effect. I think the Irish get too much credit for the spread of christianity. I would give my dues to the Franks.
The Irish, or rather the spiritual descendants of the Celtic Church were better persuaders. We see this with Norse Kings preferring English clergy to those from Hamburg-Bremen who, among other things were seen as stooges for the Holy Roman Empire, while the mostly Danelaw descended priests were trained in monastaries founded by Irishmen
And yes, I am speaking broadly and and in general terms.
The primary legacy of the Franks and later the Holy Roman Empire is the military threat going back to the Saxon Wars or before and going straight fhrough to the Northern Crusades. A threat that can also vanish if they accept Christianity. It's a tradition of people like Olaf Trygvasson, who years ago I remember described as "Christ's best hatchet man".
And yeah, I think the economic/military threat had more to do with the spread of Christianity than anything else. Based off experiences in similar conversations that will piss someone off. It doesn't matter for this argument.
I would suggest, without the 'cultural translation' that has a lot of roots in the Irish Church, Western Christianity doesn't get as far. We see this in the Orthodox Churches that are generally in former Byzantine areas and Russian or formerly Russian dominated areas and has little outside influence. Just compare the worldwide influence of the Bishop of Rome and the Metropolitan of Moscow.
Simultaneously, without the military threat, say an aborted Frankish Empire, your more likely to have more situations like the Anglo Saxon conquest of England, where the grandchildren of Romanized Britons are Germanic Pagans. Even the Christian wives of Pagan rulers aren't taken as seriously in spreading their faith without daddy's relatively bigger army in the back of their husband's minds.
Reduce either of these factors, and you're likely to have a less influential Christian cocktail and more likely to have a surviving Germanic/Norse Paganism.