I took a heavy with natasha as an engineer, within flame thrower range about and only the shotgun. The amount of health i made it out with was low, enough that normal sasha would have killed me. The heavy is about killing and thats the problem with natasha.
Sure on paper natasha looks great. Makes him more useful longer range and should hit more because of slow at medium/close. Since a heavy isn't going to walk around with his gun down, theres not point to being good long range, the 3-4 seconds it takes to actually hit the guy and either party is usually dead. Let alone the time you lost because you stopped to shoot. If something is close range, you want it dead asap. You're a big fat slow target, so sasha is the choice. Natasha only has a chance at medium range. That is where it shines the most, but the problem is that the target gets another shot or two to kill you. While you have more hp, it is not enough to cover for the lack of damage. Instead of a soldier getting 2 rockets off, he can get 3-4. Or as i did with engineer, unload two more shotgun shots into him. The heavy is fat so theres a good chance of all bullets hitting him. That 25% less damage is 2 seconds longer i get to live, thats two more shots and about 160 more damage. No class can take that type of punishment.
It also has the wrong role attached to it. You don't want to be a target with natasha, if someone runs at you they don't care if they're slow (aside from pyro, scout usually closes the distance by the time you're ready to shoot him) But you will become a target, with less killing power. All that means is you have to count on your HP more, which is a horrible trade off. Theres alot more to this than most people think, and right now i'm all over the place with it. It comes down to a killing class getting less power to slow someone who has a ranged weapon. Getting your medic killed and then killing the enemy is damn good trade for the other team.
In the end it comes down to not testing it competitively, give the gun to anyone in the competitive mind set and have them use it in the game, it just doesn't work in practice, but it is one hell of a paper idea.