I come from the south of England and say "baahhth". My flatmate comes from the Midlands and says "bath" with the same "a" sound as in "sat". There are dozens of words which fall into this category. It's not even as simple as a north/south divide. The Midlands are, predictably, in the middle of the country. People from the West country (Devon, Cornwall, Dorset) use a pronunciation which is halfway between the south/Home Counties pronunciation and the Midlands/North pronunciation. In Scotland, it's generally the short clipped "a" (from "sat"). In Wales, it's more like the Southern "baath" but softer.
I could go on for hours coming up with various regional accents and their many and varied ways of saying bath/grass/path etc. It's one of those things that you have to accept - there is no "right" way of pronouncing it and there is no such thing as an "English" or "British" accent.