Why would a car shake after new winter tires installed?

Couple of things to check... First, make absolutely certain that the lug nuts are tight. Perhaps they didn't get the lug nuts torqued up.

Second, have the tires balanced. If both of those dont help then it is a possibility that one of the tires is defective.

Ans 2. Wrenchdude knows what he is talking about; every single thing he says above is correct. Are you frightened by what he wrote ? You should be. It seems rather likely that the wheels were not balanced. Whenever you change the tires on the front wheels you must have them balanced; if you didn't you should do it now.

If you ordered and paid for balancing you can easily check. Look at the front wheels just where the tires meet the rims. You should see a small, shiny lead weight clipped to the rim. If you don't see any then the wheel wasn't balanced. If you do see them then you still need to check.

Drive along a good straight road in light traffic (better still no traffic). Slowly increase speed from 30mph to 70mph, holding the wheel lightly. If you have unbalanced wheels on the front, the car will start to shake/judder at a particular speed (50mph is common), and you will feel a bit of shaking on the steering wheel. You slow down, it goes away; speed up, it comes back.

The fact that you have fitted winter tires indicates that you probably have snow and ice where you live. Do the above test on a well plowed and sanded road, but do it. If you get a shake while on a slick road you could lose control.