No offence intended, I just hadn't read one word saying your quail were actually broody - and you might not believe it, but sometimes people actually come here and ask 'do they have to sit on the eggs to make them hatch?'. Still, am I thinking I read something I didn't or don't you only have 2 hens? If you are getting 1-2 eggs and only have 2 hens, either one of your hens is very productive or the supposedly broody one is still laying..
Anyway, for the cold I wouldn't worry too much, but the rain could be a problem as your cage looks relatively unprotected. You might want to consider covering it on 2-3 sides to keep the inside dry. Depending on how the other quail respond to the chick, you might need to separate mother and chick anyway though. And in that case, you might as well place them in a cage inside..
As for checking on it, sometimes everything is fine even if you don't check on it at all apart from when you are feeding it and sometimes you can spend 3 hours staring at it, leave for 20 minutes to get something to eat and come back to find a cold chick, stretched out on its side and about to die..