Excellent link.
But it still leaves a huge area of uncertainty.
Look at Sheila's Wheels, Esure, Sainsbury's and a couple more, there is a different notification requirement if the car has alloy wheels, or they are to be fitted. That should cover most of us.
And more importantly, in my opinion, is that the article only deals with fitting winter tyres, as in the black things that go round the wheels, and makes no reference to changing the wheels themselves.
So, if you have a spare set of wheels for the winter tyres, rather than removing the summer tyres, then you are changing wheels, regardless of what tyres are on them.
So it would be interesting to see a similar article which asks the question "do you need to be advised if different wheels are fitted?". I would expect a different table, as insurers tend to consider wheels (which admittedly are usually upgrades) as modifications to the vehicle, and wish to know.
A lot is hidden in the "Requirement to disclose material information ..........during the life of the policy" clause, quoting from a motor insurance policy wording from a major insurer.
So I'd say, for the cost of a phone call, it's best to check.