There are no gloves for children younger than 4 that will keep their hands warm and dry enough while simultaneously allowing them the use of their fingers. It is contrary to the laws of physics. If you put mittens or thick gloves on your younger-than-four year old, she will take them off so she can use her fingers. If you put thin gloves on, she may leave them on, but the gloves will quickly be soaked through and useless. Also, the chances your small child will be warm enough in an article of snow clothing is inversely proportional to the chances that your small child will be able to move in it.
If your small child wants to play in the snow, you will spend 15 minutes getting your child ready to go out in the snow. Your child will play happily in the snow for 15-20 minutes and then will come back in, wet, and demanding hot chocolate. Your house will be draped in wet snow clothes. Half an hour later, your child will want to go back outside, and her clothes will not be dry yet.
In spite of all this, you and your small child will both derive immense joy from her going out to play in the snow everyday.
Completely unrelated to snow play: small children are able to fit themselves very nicely into bathroom sinks.