My DCs all left for school sporting an Irish accent (a bit diminished over the years but still distinct from the Chicago accent around them thank God ) and those that learned in school came back just fine. A south Dublin Irish accent may be closer to American than British English, but the DCs didn't have an issue, though some swore they sounded Irish. I never made any remarks at all about their American way of saying things even if some pronunciations grated on my ears. I had a relative who used to be very pass remarkey when I was growing up, and I think it made me self conscious about my appearance and the way I spoke.
The accent isn't going to be an issue as far as reading goes. Actually, if you think about it, while SE British pronunciation may make perfect sense to you, it isn't phonetic. Your use of the letter R in your example illustrates that. Your child will hear more of an Irish or Scottish R in school, or more of an American R since the school is in America - that is actually a pronunciation that lends itself better to phonetics.
Two of my DCs learned to read before ever being exposed to phonics, at age 3-4 from being read to alone, so pretty much by sight. The remaining three of them learned some time in the course of KDG before getting to the end of the KDG phonics course. DD3 said one day 'The letters disappeared!' and she could read. KDG phonics wasn't much like Jolly Phonics, etc. Yet class after class learned to read and read well. Many of their KDG schoolmates went on to great universities.
The method the DCs' school used was phonics (againl not like Jolly Phonics) plus Dolch words (or sight words) in a systematically graduated scheme. The Dolch words comprise approximately 200 words that make up 75%ish of the words a child will encounter in literature aimed at children up to about age 8/9. Mastering the Dolch words opens up the world of books to children.
Once you go beyond the basics in phonics you are basically teaching spelling/word recognition anyway, as there are so many exceptions to rules that they make the rules ridiculous in many cases.
Agree with 'ell, yeh' -- replacing letter names with sounds still gives a child something to learn and remember. Children can distinguish between cats and dogs and squirrels and they can distinguish letter names from letter sounds. It doesn't confuse. There is a lot of hype about pure systematic phonics that is pure marketing. I would guess that a lot of children know the alphabet song by the time they get to Reception in the UK, and know which letter is which, but they learn phonics all the same.
Trust the US teachers and read to your child at home in your own accent. The brains of five year olds are very plastic.