This is a fun question. Probably there is no practical implication of this since it is “water under the bridge” as they say, but it does raise a potentially interesting question–one which just might have an equivalent today (for example if we were commanded to serve in a war but had a Christian conviction that this is sinful).
My answer is that this is a grey area. But then again, the fact that the colonists rebelled against the king raises even larger questions about Christian ethics than the paying of taxes.
I am personally very uncomfortable with situational ethics in a Christian context. For this reason, I would suggest that a Christian ought to pay his or her taxes unless there are absolutely overwhelming reasons not to do so. Christians, by nature, are not particularly likely to take part in a violent rebellion or to rebel against the leaders of their country in general. I believe that there are exceptions to this because it is always possible that the actions of a particular government are so outrageously egregious that a Christian simply cannot participate as a matter of conscience. It is conceivable that a Christian living in Nazi Germany, when learning that the taxes were being used to commit genocide on the Jews on a massive scale, might choose to express his/her horror by refusing to pay the taxes.
To be honest, the situation of the Americans under the British was unjust, but not extremely egregious. Probably the situation in Rome in the first century was at least as unjust, yet Jesus commanded that we pay our taxes. I would imagine that if I had been living in the Americas just before the revolution, I would have continued to pay my taxes. All this would have eventually become confusing, because the American rebels gained the upper hand and formed their own government which raised its own taxes. Probably there would have been a transitional period in which it was not clear what the Christian thing would be to do. In the year 1776, who should I pay my taxes to? At some point, a Christian would need to pay taxes, and would choose base on practical considerations, plus perhaps a judgment about which was the most just side in the conflict. If you lived in Syria today, would you pay taxes to Assad? Would you join the rebels? Would where you happened to live affect your answer? These questions defy simple answer, but the principle remains. Christians should avoid violent rebellion and, unless given overwhelming reason, should pay his/her taxes.
Should a Christian have joined in the rebellion against the legitimate rulers–the British and their king? I say that Christians should not take up arms in general and that “our weapons are not those of the world.” Christians should not count on either politics or military action to achieve righteousness. I am afraid that if the colonies had truly been Christian, most likely the US would not have gained independence. For all I know, we might still have been under England to this very day (although I doubt that very much).