Accent

There isn't a singular British accent that they could all have started with anyway. Britain has many very different accents. Unless everyone just stuck with people from their own area and accent, I'm sure everyone started picking up aspects of other accents on the boat over, and a new accent started coming to life then.

Then add to that the fact that America also had millions of people from many other countries, all with their own national and regional accents, it's not surprising that people there do not speak with British accents.

If anything, British accents have been merging and becoming less diverse as time passes. The invention of the printing press did much to start some standardisation of English. There were certainly many different accents, and actual dialects, at the time of the first travels to America. You can find lots of examples of early travellers around Britain being unable to understand the different versions of English being spoken in different areas and the confusion resulting from this. Some areas of Britain have their own languages (and some have died out, for example Cornish, although there are people trying to resurrect it).