Basically, we know that—despite the History Channel version of Vikings—there were actually a lot of contacts between Scandinavia and the British Isles before the Viking Age. For example, the rich finds at Sutton Hoo which are frequently associated with Beowulf actually had many similarities to other artifacts found in Sweden from the 600s and 700s. During the same period, the churches in the British Isles were developing their own art styles and spreading them to the Continent and even up into Scandinavia as part of their missionary work. This complex history of migrations and influences overlay an even deeper history of shared 'Germanic' art styles, which informed some of the underlying similarities between early medieval traditions of art in both the British Isles and in Scandinavia. (This included a common emphasis on animal elements and interwoven designs—part of the reason why some tattoo enthusiasts have a hard time distinguishing 'Celtic' from 'Viking' motifs.) In short, I see the Viking Age more as a small part of long-term cultural interactions (including migrations) rather than as a single moment of cultural transfer.
That's not to say that there wasn't a distinctly 'Norse' style for all sorts of things during the Viking Age. It's my general impression that, particularly after viking activity increased in England in the 860s, there were increasing stylistic divergences, with some people making choices about the ways they dressed and the monuments they built that looked more 'Norse', while others made choices that looked more 'Anglo-Saxon'. Again, this wasn't so much an effect of cultural transfer. It was more about some people in the British Isles keeping up with artistic developments happening across the North Sea, while others interacted more with artistic developments happening in the courts of Anglo-Saxon and Frankish kings.
This led to some weird mixes of styles that ideas of 'cultural transfer' have difficulty explaining, such as the Manx runestones which drew on both old traditions of Irish standing crosses and the developing tradition of runestones in Scandinavia. Judith Jesch has been one of the leading scholars advocating that we understand the Viking Age as a period of diaspora. I see this as a particularly useful model, since it allows us to move beyond old-fashioned ideas of diffusion to looking at how people were actively appropriating and adapting cultural elements to suit their own needs, which is sometimes referred to as a process of creolization.
I must admit that art styles are not my specialty, and I could certainly know more about Ireland in the Viking Age. However, I can point you to a very good set of data sheets that go into greater depth, prepared by Jane Kershaw. You can read these on her page at academia.edu, but to download, you'll need to sign up for a (free) account.
For a more thorough overview of Viking Age art, see the comprehensive and lavishly illustrated James Graham-Campbell, Viking Art (2013).