At least in wife's case there was an additional 4 month gap between "starting treatment" and actual surgery.

And with expected times like that, you are entirely free to use a private hospital/clinic. Both myself and my wife have done so twice.

for certain conditions 3 months wait can be very serious.

Yes, but this is a general trentment gurantee. If you get cancer the times have to be shorter, and are shorter usually messured in weeks. It's better but still not good. If there is suspected cancer, you should get diagnosis started the same or next day, and treatment/surgery within a week.