2014 LandCruiser Prado: Wobble in steering wheel.

I’m inclined to think your tyre store might be right, Anthony, and heavy vehicles like the Prado, especially if they’ve been used of-road can, in fact, damage their own rims to the point where the wheels aren’t perfectly round any longer. It would have been pretty obvious to the people fitting the tyres when the new tyres were balanced; the out-of-round rims would have made balancing them almost impossible. Don’t be fooled by the fact that the wobble happens at speed, because that’s simply a function of the harmonics in the wonky wheels building up to the point where you can feel it.

However, to answer your second question, yes, there are some other things to check before rushing out and buying new rims. Is the vehicle lifted on its suspension? If so, there’s a very good chance the wheel alignment settings (particularly the castor angle) can be thrown out. This needs to be corrected to maintain the vehicle’s stability at speed, and a wheel-wobble can sometimes show up as a symptom.

How are the suspension linkages and bushes generally? Worn bushes can easily cause a vehicle to wobble at speed, as can bent bits and pieces from off-road use. Another thing that can throw a wobble into the equation is an out-of-balance tail-shaft or worn CV joints and universal joints in the driveline (of which there is plenty in an all-wheel-drive Prado).

But if it is your wheel rims at fault, there’s probably not much else you can do but replace them. Sometimes a steel rim can be re-rolled or trued, but alloy wheels (which is probably what you have) are often throwaways once they’re bent out of shape.