Westminster conciliates the US colonies by giving them seats in Parliament. As British North America expands westwards, it grows in population and wealth, until eventually its representatives make up a majority of MPs (and possibly Peers as well). They decide that they don't like having to traverse the Atlantic to get to their legislature, and vote to move the capital to somewhere in BNA, with London remaining the ceremonial capital for coronations etc. There's a lot of grumbling in Britain proper, but people ultimately come to accept the new reality.
As a slight alternative to the above, have Westminster conciliate the colonies by setting up an Imperial Parliament, with both British and American (and later other colonial) representatives, to deal with foreign policy/defence, whilst Westminster itself and the various colonial assemblies deal with internal affairs in their respective countries. The American representatives might vote to move the IP to BNA, or they might not (if it scrupulously keeps its nose out of colonies' internal affairs, there'd probably be less concern about its location); even if they don't, North America would still be the dominant part (in terms of population, economy, etc.) of the Empire.