What a snowing day!

[Silverobama]

I woke up seeing it snowed heavily today.

Is it natural to say "What a snowing day!"?

 

[PeterCW]

No, we would refer to "a snowy day".

 

[teechar]

I woke up seeing it snowed heavily today.

That's no good.

You may have come across something like:
I woke up (still) feeling tired.
In that kind of sentence, there is some logical connection between the two actions. In the one you wrote, there isn't.

Is it natural to say "What a snowing day!"?

No. See above. Again, you may have come across, for example:
What a great day!
What a lucky day!
or
What an exciting day!
The pattern here is: What + a/an adjective + day!
However, "snowing" is not an adjective, and even if you weren't sure about it, here's a little test you can do. Ask yourself if you can use the word "very" before it. If you can't, then you can't use that word in the above construction.
Thus:
What an interesting day!
What a long day !
What a snowing day!