If you look at the fare breakdown (the option to show this should be under the section that shows "Your total price = $/£/€") you will probably see that the prices you are seeing initially are just the fare element, not including BA's carrier surcharges and taxes. I'm not sure, but I'm guessing that unlike requirements to show the full cost up-front in say Europe, there may not be the same requirements to show this in all countries.
It's the US based BA.com site, although same happens on MX based site. US BA always shows US full fare price in initial selection portion for US to Europe fares.