Given a circular wire connected to a battery with a potential difference of v across the wire. And given the fact that resistance in the wire is zero. If a constant potential difference is applied across the wire then the electric field does work on the electrons causing them to accelerate. In a wire with resistance these electrons move at a constant speed even though under acceleration due to resistance bein proportional to electron velocity. Now the problem for me is when r is zero each electron is under uniform acceleration in the wire, meaning they are getting faster and faster. So does this mean that over time current increases and increases for the same potential difference? And if so if p=IV doesn’t this mean p increases for the same potential difference? So you could just wait for electrons to accelerate theoretically near the speed of light?
The potential difference V between two points is defined as the work required per unit charge to move the charge between the two points. In the theoretical case of a zero resistance wire, no work is required to move the charge between any two points of the wire, so the pd would be theoretically zero all along the wire.
As far as connecting a zero resistance wire to a battery is concerned, all real batteries have internal resistance. Call it $R_b$. The batteries has emf is its terminal voltage without anything connected to the terminals. See top diagram below. If a zero resistance wire is connected to such a battery, the terminal voltage will be zero and all the battery emf will be across its internal resistance, $R_b$ per Kirchhoff's voltage law. See bottom diagram below.
Of course all real wires (except superconductors) have resistance as well as all real batteries have internal resistance.
Hope this helps.