I've heard it referred to as the Mission Twang. My knowledge of it is strictly anecdotal, but I think it was an accent most widely found among the MissionIrish (traditionally spoken as one word). There are pronunciations common to both the Mission Twang and Irish such as sometimes vocalizing a final "t" as "ch" as in "I'll meetcha tomorrow." and substituting "youse" for "you".
I pronounce "wash" as "warsh" although I never lived in the Mission.
I don't think anyone under about 70 has the accent anymore.