Languages evolve. The English language spoken in the US may have had its roots in England, but the English spoken here has no relation at all to the English spoken in England. In particular, the 'slang' terminology differs greatly from country to country, sometimes from one generation to the next in the same region. A century ago, being gay meant 'to be happy'. Tell someone you're gay in the US now and the first thought of the other person isn't that 'you're happy'.
Nor do the people speaking English in Wales, Ireland, Scotland, India, South Africa or Australia sound anything like an American, or even an English person. Even different areas of the US sound different from each other. A Texan dialect doesn't sound anything like a New York one. And to a trained ear, a Texan doesn't sound like people speaking English who hail from Georgia or Tennesee either.
Dialects are picked up within the first few years by infants from the adults in their sphere, one reason you shouldn't speak 'baby talk' to them. They learn to speak their language properly by emulating their adults. Also many people are good with language/dialects, and tend to pick them up quickly, but a native can usually tell the difference in someone adopting a dialect and someone who's actually been speaking it for many years. When I lived in Chicago for a couple of years, I adopted the same midwestern dialect people around me were speaking, but I lost it quickly when I returned home to the southeast. I shocked a friend on Skype when i imitated his English accent, though he knew it wasn't my usual speaking voice (I have a decided southern twang to my voice when i speak normally).
The elves lost not just their culture, but their immortality and language a thousand plus years ago. They won't sound like the elves of 1000 years ago, and shouldn't. Doubtless their language should be bastardized by the environment, new speakers, new phrases and words that come up, since the original speakers of the undistilled language from long ago are long dead (with a few exceptions). Even the "elvish" spoken by the Dalish (who are the only modern 'experts' in it, such as they are) is mostly a few scattered phrases, with the majority of them being used for ritual purposes (such as the DA2 clip shown). Or simple greetings/titles. I've seen little evidence that shows they speak more than a few scattered words here and there and mostly the prevailing human language of the region they grew up in.
It wouldn't surprise me if some Dalish speak with a 'Welsh, Irish, or even Scottish or English' accents since this region we're playing in corresponds to that real world equivalent environment. The city elves (and Dalish of DAO) spoke with American accents and I found the switch a bit jarring in DA2. As with the physical changes, though I just shrugged and went with it. As a long time reader of comics and watcher of daytime dramas on tv, frequent artist changes and actor changes to long term ongoing stories came quite easy to accept for me.
I am unsure what other accents are employed elsewhere in Thedas. Antiva seems to be Italian or Spanish based, but Rivain doesn't stand out to me as anything significantly impacted by any real world equivalent (or at least Isabela of DA2 doesn't employ an accent that I can tell). Orlais is French, but again, the accent varies. Leliana doesn't have the same accent as Riordan or Stroud--I only bring up those two examples since the actors for the characters are both French speaking natives, but it illustrates a range of regional/environmental differences in language. Corinne Kempa studied in England so she has picked up something of an English accent to her voice, though it's still distinctly French. Nevarra may be English based as well (since Miranda Raison is an English actress), or Cassandra may have picked up the 'Ferelden accent' because she's spent significant time in the region. I would expect in other areas where other languages are employed to have an impact on the elven dialects as well and I'm actually kind of curious what we might get from elves in Tevinter or Antiva.
To have anyone in Thedas employing an American accent sounds weird to me because regionally it doesn't make sense. Dwarves, sure, they're an isolated group. Qunari even, since they are originally also from another region. I would expect elves who live around dwarves or qunari in their home turf will adopt those accents to their elvish as well. With possibly a 'lilt' of something a little more that marks the elvish as a different language.