I'm on my work PC, but I can tell you a few things.
1. I do have an ATI card in my home PC.
2. It's happened to me at least a dozen times.
3. Even if you close LoL, your cursor will be screwed up. That's why I restart my computer instead of restarting LoL.
4. Sometimes it randomly goes away, if you wait. Could be 10 seconds. Could be a minute or two.
5. You can see the cursor (scribbly lines) when you cursor over something, but you can't see it until then. So it's pretty hard to buy stuff, fight, etc. Capping is the easiest thing to do, because the box is so large. Heh.
I've tried various solutions to no avail. The problem is you can't predict when it's going to happen. So every solution will work...until it doesn't. By the time it happens again, so much has changed (settings, updates, etc) that you have no idea what in particular could have caused it.
It's very hard to pin this bug down. If I could reproduce it, I would, but I'd also probably be able to avoid it. Like if tabbing to Firefox caused it, I'd just not tab to Firefox during a match; therefore, this wouldn't be much of a problem. Unfortunately, it is a problem, and it's very displeasing to lose a fight and/or a match when it happens. Your only hope is to restart, get a lucky fix within 10 seconds, or hug a turret and hope people overestimate your ability to defend.