If you put cold milk in a thermos how long is it good for?
Depends how cold it stays. It won’t be sterile, especially once decanted into another container. So once it hits anything warmer than about 40 or 50 degrees, we’re talking a matter of a few hours.
It’s not sterile in the original container either. It’s pasteurized, meaning most of the harmful bacteria are killed. Sterile means all bacteria are killed. For a simple at home demonstration of the difference, leave a sealed gallon of milk out in the sun for a few days then take a sip.
(There are also a couple of types of “super-pasteurized” milk that’s closer to being sterile, but I assume the OP is talking about regular old pasteurized milk)
That’s why I said “especially,” not “at least.” 
I was emphasizing that already non-sterile milk was probably picking up additional bacteria from its environment, in order to explain why it would start going bad as soon as it became warm.
If you can keep the thermos in a cooler with ice it should last a long time.
Fresh (unpasteurized) *breastmilk *can be kept in a cooler with ice for up to 24 hours.
For what that’s worth.
Smooth. Now all I need is a plunger, some duct tape and a woman who needs $20.