I didn't think there was an exemption for classics.
I know some of the London to Brighton runs have had to start outside the ULEZ due to charges.
Copied from Tfl site:
Historic vehicles
You can apply to stop paying vehicle tax if your vehicle was built more than 40 years ago. This date moves forward on a 40-year rolling system. For example, when the ULEZ launched in April 2019, vehicles built before 1979 were eligible to apply for historic vehicle tax class.
All vehicles that have a historic vehicle tax class will be exempt from the ULEZ. This tax class excludes any vehicle used commercially (for example, coffee vans or street food vans).
In line with the existing LEZ discount all vehicles constructed before 1 January 1973 will be exempt from the ULEZ, regardless of commercial use or otherwise.