Above image from flightrader.com. It's going up fast at the moment - something like 1000 feet per minute. Looks like it could be headed to fly over Timaru in an hour or so, but despite being large, I guess it may be no more than a very small dot in the sky. I think maximum height is over 100,000 feet.
Actually already over 70,000 feet while I typed this. I guess they'll be very happy with that.
Edit - it's not heading for Timaru. At about the 70,000 foot mark it climbed out of the SW jetstream into an Easterly, so reversed direction and is now heading to Australia at about 20 knots. Earth.Nullschool.net data suggests that if it stays at that altitude, it''ll head over Melbourne, then loop back around south of Tasmania, and might end up back over NZ.
Another edit - Flightradar folks have lost the signal. It can still be tracked on NASA site. It's already at 105,000 feet, heading NNW at 50 knots, and I can't see predicted winds at lower than 10hPa on nullschool, so no idea where it's heading.