i think your mum's right on this one. Plenty of girls experience breast itchiness and discomfort when they're growing.
Mine never really grew that much, so I can't speak from personal experience, but it's well-known that this is a common occurrence during puberty!
What To Do When Your Breasts Are Growing
First of all, stop worrying — it sounds as if there's absolutely nothing wrong.
Secondly, make sure you keep them moisturised. Cocoa butter, or anything with Vitamin E in is a good idea. Otherwise, your breasts could have a sudden spurt of growth, and you could end up with stretch-marks on them.
If your breasts are growing, chances are, your hips and thighs will do the same. Again, moisturise with cocoa butter to prevent stretch-marks.
I don't know what to suggest to help with the itchiness, unfortunately, other than make sure they're comfortable. Also, sleep with your bra off. When I was about twelve, all the girls were saying that you had to sleep with your bra on so your boobs wouldn't sag. This isn't true.
Sleeping with your bra off is a lot more comfortable, won't affect the shape of your boobs, and may even stop them itching in the warmer months.