Does the wire gauge affect RF emissions when there is a relatively high current?
I was wondering if the AC current carrying wires were to experience
high current if that could cause RF emissions of significance. If so
would increasing the thickness of the wire help to reduce emissions?

If the RMS AC current is high frequency then it could be regarded as a source of RF emissions. So this is my assumption in answering your sub-question and, of course, if the RMS value increases, then it becomes more of nuisance RF emission source.
But, would increasing the wire diameter abate this noise source; would a thick wire with a naturally reduced current density (compared to a thinner wire carrying the same current) be less of an emitter of RF noise?
The simple answer is no. If you analysed the magnetic field generated at some distance Y from the dead centre of a wire carry X amps, it would be the same irrespective of the diameter of the wire.
We can start to see this in a coaxial cable. We know that if the forward current through the (much thinner) centre wire is the same magnitude as the current returning in the shield, then the external magnetic fields cancel out completely. So we have (a) a thin wire carrying X amps being compared to the shield (a much larger diameter) tube carrying -X amps. They produce the same external magnetic flux density but of opposite polarity and they cancel.
Scenario 1 - B field around a shield carrying 1000 amps

Scenario 2 - B field around a shield with inner carrying 1000 amps

As you might be able to see (in the corners) the B field values (colours) are the same in both scenarios. Images courtesy of QuickField student edition (a nice little free tool unfortunately limited in finite element analysis nodes but usable in this situation).
It's not much of a step to realize that this would also be the case should the &quot;tube&quot; be actually solid. I have done this FEA analysis but I can't find the picture for when the shield is all copper to the centre. However, there is this picture from HyperPhysics that proclaims the external field to be unrelated to the wire thickness: -

Where there is a difference is the near field. If the thinner wire is 5 mm diameter and the fatter wire is (say) 10 mm diameter, between 5 mm and 10 mm there will be a measurable field for the thinner wire that can't be measured in the fatter wire (for obvious reasons). I'm not saying the fatter wires doesn't have that field; I'm saying that it is internal to the wire and of no interest when it comes to induction of RF noise.