Quite a few ATCOs are also pilots, but they're ATCOs first and foremost and most were pilots of some description before the before they trained to be ATCOs.
The nature of our shift systems allows for sufficiant time off to get the flying work as and when it becomes available on rest days, but whether you'd get enough time off, earn enough money and be disciplined enough to undertake an ATPL at the same time is questionable (not a personal reflection upon you, just the reallity of the situation).
I'd like to think you could do it, but it's an enormous task, some may say bordering on the unachievable and you could end up meeting yourself coming back if you're not careful.