My DS is 1 week off his 2nd birthday and has started to not sleep during his lunch nap.
I still put him down for an hour of quiet time in his room between 1:30 - 2:30 but he just rolls around chatting to himself rather than falling asleep. He's full of beans in the morning and after lunch and sleeps solidly for 12 hours at night so I don't think it's doing him any harm not napping - I do an earlier bedtime for him so he gets a bit more night sleep on the days he doesn't nap. He dropped his late afternoon nap pretty early on and can't nap in his pram as he gets major FOMO so I'm not completely surprised he's decided to stop napping. I'll keep offering it for a few more weeks incase this is a sleep regression or he still needs the awake but quiet time in his cot to rest.
If your child dropped the nap at 2, what did that look like for you? How long did you keep offering it for before you gave up? Do you do something restful at nap time instead to have some chillout time?