Odd question, I know, but in the rare images I've seen of Britons fighting invading Norsemen, the Welsh wielding a two-handed axe is persistent, at least enough for me to pick up on it. Was this a common thing? I'd always associated them with the huscarls and Scandinavians of later years, but these are depictions of the early invasions, when the Sons of Ragnar were still running amok. Follow up, if this was a thing, did the practice extend into the Hen Ogledd as well, or was it primarily a Gwynned/Glywissing/all the others in that area thing?
If you don't mind linking some of said images, that would be very helpful in determining their historicity.
That being said, we have very VERY little information about the specific armaments of soldiers in the early Viking period. For example, Ragnar Lothbrok was himself a semi-legendary figure who we don't even know existed. The idea that the invaders of Britain/Ireland in the late 9th Century were definitely his sons is not even 100% certain. This is just to give you an idea to how limited our knowledge is of the period.
What's more, the best primary document is the Anglo-Saxon chronicle for this period/place. As a result, it's primary focus is on the Anglo-Saxon side of the struggle with the Danes. This means that our knowledge of Welsh conflict is even less detailed than what little we have to go on for the English kingdoms.
All that being said, the Axe and Spear were far more common than the sword in general for the medieval period. Thus it is more than possible that Welshmen would be armed with axes of some sort to combat their enemies. But the large, two handed 'dane axe' was specifically known as a Scandinavian weapon, at least until later than the period you are referencing. It's again possible that some Welsh would use it if they found one, but my money (and we are dipping very far into hypotheticals) would be on the usage of Spears. This is partly from analysis of later Welsh armies - for instance, about 400 years later we have the Welsh forces facing Edward I primarily armed with spears. For that matter, the majority of English, Scandinavian, Irish, what have you would have primarily used the Spear.
Also the depicted use of the Dane-Axe by huscarls is very much a result of their depiction in the Bayeux Tapestry, one of the few extant visual displays of medieval battle. But the Tapestry was made nearly 200 years after the time period you are looking at. Part of the problem with modern recreation/depiction of the viking period is that they (perhaps unknowingly) use the Tapestry as a primary source for viking/saxon/etc. weaponry, when really it might have been completely different. We just don't know.
Here's some further reading you might like to do. Please link the images if you get a chance!
For primary sources you could look at the Anglo-Saxon chronicle which at times discusses Thegns, Huscarls as well as describes some tactics etc. This can be found relatively easily online in multiple translations. Here is a website that breaks down the chronicle by years. http://www.britannia.com/history/docs/asintro2.html
Also a glance at the Bayeux tapestry would give you a vivid idea of what contemporary huscarls would look like as well as their equipment, armaments etc. Here is a website that breaks down the tapestry by scene, and I have linked specifically a scene that shows the English Infantry. http://www.bayeuxtapestry.org.uk/Bayeux27.htm
Lavelle, Ryan. Alfred's Wars Sources and Interpretations of Anglo-Saxon Warfare in the Viking Age.
Oakeshott, R.E. (1996). The Archaeology of Weapons, Arms and Armour from Prehistory to[ the Age of Chivalry.