In the English I speak, we would not use the word tissue.
We'd say "Get a kleenex" or "Get some kleenex."
Kleenex is actually a specific brand, and the manufacturers are fighting a losing battle to stop people from using the word in this way. They want us to acknowledge that it's a brand name by spelling "Kleenex" with a capital K.
If a newspaper printed "Get some kleenex" they'd get a nasty letter from the Kleenex legal department. There's info on this product at kleenex.com.
There are other examples of a brand name being applied to the product in general. "Aspirin" is another. And "Walkman."
Her nose is bleeding. Get a tissue!
Would it sound more natural to say "Get a piece of tissue" instead of "Get a tissue?" Thanks.