Yup, bree most of the time.
And Edinburgh, at least where I live, is not pronounced “EdinbUrrow” as - in my limited experience, Americans like to say - but “Edinbruh”. Although I’ve been told that the “burrow” pronunciation does exist in some places in the UK, I’ve never heard anything but “bruh”.
As far as I cna tell form watching US TV, Americans tend to pronounce all the syllables clearly in a word (especially those ending in “ory” and the like, which is very pronounced), and will emphasize lesser words (conjunctives, things like “them” and stuff, whatever they’re called) clearly. So they’ll say “I gave it to them” with the “eh” sound in “them” very clear, while Brit treats the “them” as not much more than a grunt; “thm”.