Rope was around long before the wick. It seems conceivable that rope shavings or pieces or old rope were an easy way to start a fire.

This was then used with oil to make an even better fire starter or means of transferring fire. Eventually someone realises that a rope soaking in oil is easily lit and sustains a flame.

Before wicks, how do you burn oil? It's not easy to just ignite a bucket of lamp oil (putting aside what you might make the bucket out of). Probably you soak other fuel like wood or rags in the oil and burn the result. It's not a huge step from there to accidentally find out that you can make do with one piece of wood or cloth or string for a lot of oil, assuming you have something to put the oil in.

1. Someone dips a rope in flammable oil before lighting it, and sees that it's quite flammable. 2. Some other time, someone tries to use a rope dipped in flammable oil as a fuse to trigger the lighting of the oil once it burns back to the oil 3. They notice that the fuse keeps burning but doesn't burn back - in other words, the wicking effect 4. They shorten the rope and reshape the pot, and that's an oil lamp.

The first oil-lamps were basically a bowl with animal-fat and plant parts as wick. It's not hard to imagine how ten thousands years ago a hunting tribe could discover such a device by accident or on purpose.