I am Italian. My company has an office in England (South East area) and one in LA. I got pretty good at picking up UK accents, not so good with the American ones.
In this game, I prefer the british voice actors to the american ones, but it has to do more with pitch than with the accent itself. I think the american ones are too deep, and I kind of see them as suited for Qunari only.
The british accent doesn't sound too posh to me - actually, I am happy they went for a slightly posh BBC accent: I think it fits very well.
To people complaining about other accents, I was surprised about this. I thought Solas Welsh accent and overall voice acting was beautiful.
EDIT: hopefully my statement above doesn't sound creepily fan-girlish to Welsh people: I think the mix between accent, delivery, and the way the lines are written is very musical - that's it.
By the way, I got a little traumatized in my first playthrough as I had the "customized Hawke" bug and my character's voice chainged abruptly from the English one to the US one. I had to roll back to a previous save and picked a "standard Hawke" to keep the voice I had chosen for my Inquisitor.