My 2.5 year old goes to gymnastics and loves it! It is a "real" gymnastics class - they have a little balance beam, bars and a trampoline. Their classes start at one year old (I know it might be just a little far for you to travel (), but here is the website in case you're interested.)
She's definitely old enough, but I can't help you with where to look - sorry!
Try to find a 'real' class. DS goes to Crystal Palace National Sports Centre and is taught by professional gymnasts ... they do adult-size beams, asymetric bars, trampoline, rings (with loads of crash mats) as well as obstacle courses
My dd1 started gymnastics at the local gymnastics club when she was 2. It was baby gymnastics but they had all the proper equipment just scaled down and it was run by proper gymnastic coaches. She's 7 now and still goes although obviously has moved up to full sized equipment now, she loves it. Try your local sports centre or phone your local council or library they'll have information on these types of services in your area.
Our local(ish) leisure centre does classes leading to Persil and BAGA awards. The classes start from 18 months, though nothing very structured until age 3. DS1 (3.9) is in a class which lasts for an hour and they do beam work, vaulting onto a horse, and various floor stuff (forward/backward rolls/cartwheels). The class did a sweet little show at the end of last term.
Here they can't start 'proper' gymnastics until they are 6: It is run at the council leisure centre, but is never advertised, as it's always full with a waiting list - so you may want to ask/phone around.
Where we used to live there was an after-school club (but proper gymnastics with a qualified coach), and children could go from 4.
We have a fantastic gymnastics club a few miles from us and the children have been going since they were toddlers. DD is now on their waiting lost to do Gym Club every Saturday, tbh she would have been ready a lot younger, just that I didn't put her name down until very recently. Could you ring your local Sports Centre and ask for the name of the local Gym Club, if they don't have it, they should have a number of someone who does, if not typing Gymnastics Clubs in York on Internet should prove fruitful hopefully.