"Let every soul be subject unto the higher powers. For there is no power but of God; the powers that be are ordained of God. Whosoever, therefore, resisteth the power, resisteth the ordinance of God; and they that resist shall receive to themselves damnation."
This passage was and is the most troublesome political passage in the New Testament, and was responsible for literally millions of words exchanged on the question of political liberty. Men like John Calvin took it as an absolute prohibition against anything resembling revolution or revolt against even the meanest of rulers.
John Locke's treatment of Romans 13 is pretty straightforward: Christians are not exempt from obeying lawful authority just by virtue of being Christian. They have to obey the same laws as everybody else.
On what is "lawful authority," Locke says Paul the apostle "is wholly silent, and says nothing of it," because for Paul or Jesus "to meddle with that, would have been to decide of civil rights, contrary to the design and business of the Gospel"---which of course was the business of salvation, of preparing for the next world, not this one.
Locke notes that it was Paul's intention and prudence, that such "sauciness, sedition or treason" was, in those times of Roman "insolent and vicious" rule, a "scandal to be cautiously kept off the Christian doctrine!" [The exclamation point is Locke's.]
Founding era preacher William Ellery Channing made a similar argument about why the New Testament didn't explicitly ban slavery: "a religion, preaching freedom to the slave, would have shaken the social fabric to its foundation, and would have armed against itself the whole power of the state."
Jesus didn't preach violent revolution, that his church would be arming itself against the whole power of the state. Indeed, we recall that many were disappointed he wasn't that kind of Messiah.
Therefore, sayeth Locke, the "lawful authority" question must be decided by worldly standards, to be "determined by the laws and constitution of their country."
And so, if a legal argument to separate from Britain's constitutional monarchy could be made---and indeed the 27 grievances in the Declaration of Independence like "taxation without representation" was such an attempt---then there was no theological impediment per Romans 13 to such a separation.
Further, Locke asserts "the doctrine of Christianity was the doctrine of liberty," using for his example that Christians were "freed" from observing the "Mosaical" law.
In other words, Locke is dispensing with any supernatural argument that unlawful rulers should be obeyed because it's God's will because Romans 13 says so. According to John Locke, it doesn't.
From Locke's A paraphrase and notes on the Epistles of St. Paul to the Galatians, First and Second Corinthians, Romans, and Ephesians, p. 367
[ HT to Ben Abbott for the above citation and link.]