I'm currently working on a fantasy story inspired by the Celts. Now, this isn't the problem, it's this: while it started as Celtic, it's also been influenced by the history of the Welsh, Irish, and Scottish, which are all familiar but not quite the same. I was wondering if it's fine to pick and choose (ex. Welsh names but Irish myths, Celtic culture) without coming off as stupid.
If you are using a complete new world, you will probably get away with it. But it might be an idea to assign 'regions' and have people with welsh names be from one region etc...
What jedellion is suggesting is the right way to go, and essentially what Tolkien did. It's fine to smoosh them all together, but as a savvy writer, I would use the different regions-different dialects thing to demonstrate to potential Welsh/Irish/Scottish people that you understand they're not all one homogenous culture. You hae nae idear how ANGRY tha Scotts get wha' some wee yank suggests they're indistinguishable from tha bloody Irish!
I agree that having different regions is a fantastic way to integrate the different aspects of culture your writing with. But I think if it's taking place in a modern time, aka the present, that meshing the cultures together wouldn't necessarily seem off-put, as in our current culture a mix of cultures is more likely than not. But also, because the cultures you are speaking of are very closely related- you could blend them together in a way that respects each culture. Names, for instance, aren't nearly as important as myths. You can find similarities in many myths. Use those similarities in your story- rather than the differences. It's a good way to stay true to the cultures without blending them in a way that doesn't exactly 'fit' them individually.
The safest thing to do is never explicitly refer to any of them as Welsh, Irish, etc. Names and themes unique to each nation should also be avoided. Instead, be creative with your naming: Either use portmanteaus to hybridise concepts or just mess with phonetics.