From some of history books I have read, the St Johns current was not very swift until they dredged the river after the Civil War.
The southbank was somewhat swampy and it gradually entered the river with the main original channel closer to the northbank.
In fact the first dredging increased the current so much it caused erosion on St Johns Bluff and parts of it fell in until they installed some rip rap to stop it.
Many think this is when the earthen remains of Ft Caroline were washed away.
So its possible at the time cattle could wade through the shallow parts of the southbank and then swim the narrow channel. With the currents so slow at the time, there was little threat to losing them.
When I lived in Chicago I read that they used to bring cattle over Wolf Point on the Chicago River. If you saw Wolf Point today you would say its not possible. But the currents back then were very slow and the banks very gradual. All the modifications by man have changed it so much its hard to tell just how rural it used to be.