I remember reading/hearing somewhere that originally the British sounded like Americans do today (not that there is one American accent, but I hope you get what I mean).

Not really. Some features of older English are shared with American English, such as pronouncing r at the ends of syllables (rhoticity). Other features of RP, the standard British dialect (what's on the BBC, and if you're American probably what you think of as "British English"), are more conservative, such as the vowel system. Both of these differences started to emerge within British English before American English really developed, and the line isn't strict. There are actually rather few features that have a hard British-American divide--many of them exist in both to varying extents, or only exist in one but aren't uniform in either. You really can't say that one is generally more conservative thna the other.

See here in the FAQ.