Interesting question. I understand my membership to cover me regardless of the vehicle I'm travelling in, so I guess no extra charge. But not owning a classic car I havent read the small print and it may well ask for the vehicle to have a valid MOT which you wouldnt have.
Both the AA and RAC websites state that they provide breakdown cover for vehicles of any age, including classic cars, with no mention of any additional charges.
If you let those pesky meerkats search for breakdown insurance though, you'll be asked to enter the age of your vehicle, with any quotes resulting from your search then showing the full price:
https://www.comparethemarket.com/breakdown-cover/classic-car/