Hmmm, that's difficult to say as I have been at it for a few years now. I would say that some benefits are felt immediately, although I'm not sure that courses are always beneficial as I think there are many different forms of meditation and some people are suited to some forms more than others.
I do recall feeling my mind calm a little the first week I started meditating, which was about 5-6 years ago. Just a little bit, but a small sign as I was breathing.
I haven't always been good at keeping the practice up, and tended to meditate only when my anxiety started creeping back, but that was a mistake on my part. It's really a long term practice that benefits many areas, long term.
One other thing that I'd like to add is that it took me a long time to break bad habits, which in my case was significant and invasive bouts of anger at everything and anything that didn't fit in with my view of the world as I wanted to see it. To be fair I was involved in a ridiculous legal battle that leeched nearly £70k out of my bank account, which did not help.
I think that meditation and mindfulness are two sides of the same coin, and that if you expect to just sit down for 10 minutes a day and for everything to magically get better (which I kinda did at the start) then you're in for some disappointment.
But going back to your original questions.....
Probably 3-6 months before I stopped reacting to panic attacks. They still happened, but I stopped 'oh shit oh shit oh shitting' about them. Just let them happened, and when they had passed I carried on as before (this is significant in recovery imo).
The other thing to watch out for is setting a time for meditation and just letting the mental noise be. It's really easy to give up after a few minutes because you think 'it's not working', but as I said previously that's not the point. Eventually it does work, and you start to feel the mental calm.
Using that same worn analogy, it is like exercising to lose weight. You don't see a difference day to day, but then 2-3 months down the road when you look back, you realise how much change has occurred.