"How difficult was it in those days to travel from the city to Islington?"

Not difficult for one free to do so.

However in "those days" an apprentice would live in with his master's household and his liberty would be almost entirely dependant on the whim of his master. It would be not at all odd for the apprentice's days to be regulated through all the hours when the city gates might be open, and for the apprentice not to be allowed out at night.
The squire's daughter might not have much freedom of movement either, especially before she had come of age.
There is no need to assume that the Islington in question was any other than the hamlet just outside the city walls, and it adds some poignancy to the song that the couple were separated only by a couple of miles as effectively as if they were on different continents.

Of course, it is only a song.