I once knew someone who was a teacher on Long Island. She had a somewhat Midwesternized accent, but many of her coworkers had strong LI accents. She was preparing a lesson one day about rhyming words, and was writing pairs of words on a poster: cat/hat, mouse/house, dog/log, and so on. Another teacher looked at the poster and said, confused, “Doawg and laahg?! Those don’t rhyme!”
So yes, how phonics is taught will naturally vary depending on the local accent, and in some cases, depending on the individual teacher. But most of the time, it’s not confusing to the students, because to them, their accent isn’t “noticeable” at all; it’s just the way normal people talk.