The right to obtain your personal information hasn't changed but what is really useful from today, is that councils can no longer charge the optional up to £10 fee unless you go overboard:
and that they also have to respond within a calendar month whereas before it was 40 days.
Currently when you challenge a PCN, or a Notice to Owner, you don't know what cards the council has in its hand, as you don't know what notes the traffic warden (CEO - Civil Enforcement Officer) made. Now you can find out fairly quickly for free. If you limit your request to that information, it cannot possibly be excessive. Then you can make a more informed decision as to whether to fight or pay.
Another time when it will be handy is when you have moved and a bailiff has popped up with a warrant for a PCN you know nothing about. Then it would be reasonable to ask to see everything.
If you have a Notice to Owner but not the original PCN from the car, you might want to ask for a copy of the PCN so that you can check how long your vehicle was observed for.
To make your Subject Access Request you simply need to write a letter setting out the information you require and send your V5 registration document as that proves identity and location (the council does not know who owns the car when they issue a PCN on street) to Information Management Team, North London Business Park, Oakleigh Road South, London N11 1NP or by email to email@example.com
The above are simply Mr Mustard's suggestions as to what data you might want to ask for about yourself but don't let that fetter what you ask for.