That's the downside of a FWD car with a high maximum torque like a diesel engine: the wheels spin very easily
An you can certainly experience this on the pumpdöse VAG diesel engines. They react more violently compared to commonrail diesels.
Most current TC systems allow a certain wheelspin during a short period of time, after which they will interact. At least that's my experience. If you really take off violently TC will interact pretty fast, but if the wheels don't spin like mad it will take some time before the TC cuts in on the throttle.
As for the understeer going into a turn: if you're turning into a turn at a speed above what's physical possible, you'll get understeer, ESP or not. ESP just helps to control the skid, but I've you cross the physical limits, ESP can't do much about it. You can easily get sudden understeer in these winter conditions with an everyday family car on summer or 'all season' tires if you push a bit harder. I don’t think you’ve to get worried too much. BTW, ESP is turned on by default when the car is started.