Do you think this is too late in the year to visit?
As far as leaves are concerned, it may not be as good as this weekend, but most likely be still good enough. As far as climate is concerned, it would probably be still warm enough to get around. But be sure to bring light and warm outers that you can take on and off frequently.
from Ueno Station, to Nasushiobara Station.
If you're not renting a car anyway, have you looked into options about getting off the train at Kuroiso station? I haven't cheched the bus time table, but Kuroiso station is generally convenient in many ways while there is almost nothing around Nasu-shiobara station.
That said, why aren't you renting a car? Was everything fully booked? Rent-a-car is your best option for off-summer-season. Also, driving is much easier compared to the city.
Meanwhile, I can understand your concerns about driving if you're a short-term tourist from a foreign country. In that case, I have to say that you should limit your schedule as much as possible. Buses aren't really that convenient in Nasu.
A couple of the things that sounded interesting were the Trick Art Museum, Animal Kingdom, Safari Park, Heisei-no-mori Forest.
Hmm, very lame, I must say. These spots can be "a bonus" to your hiking if you already live in Japan, but to come all the way from another country to see Trick Art... well, everyone has their perferences. But you can find a trick-art museum in Odaiba as well.
Either way, if you're not staying overnight, you should focus on just one of those spots, because they're mostly far apart. And Nasu is a vast area, so even with a car, it would take you time to get from one place to another.
If you truly love animals, Animal Kingdom and Safari Park can be interesting. Animal Kingdom is really a vast area, and you should take time to enjoy every aspect of it. In particular, don't settle with the cats and dogs near the entrance, but go all the way to the "farm" area where you can enjoy a meadow with sheeps and other animals. But there's nothing too Japanese about it, so it might not be interesting if you're from New Zealand.
Safari Park, on the other hand, won't take much of your time. There is a in-park tour bus, and you can't get off your vehicle or go back and forth. Huge camels will most likely come up to your window.
Among your choices, Heisei no Mori is the most sofisticated. It's also the newest and the most academic. In other words, however, it may be boring if you're not interested in plants. Also, it may be too similar to your hiking course.
One idea is to visit one of the spots you've named, and then go to Kuroiso station much earlier than your scheduled train, and kill time there. Kuroiso station area is the central town for the Nasu resort. There are shops and cafes you can look into.
I don't think taxis are difficult to use as long as you write down your destination and address (for example, Animal Kingdom is called Doubutsu Oukoku in Japanese, so it's better to have the address). But you can't find many taxis on the streets, so ask shops and clerks for taxis. Taxis can cost you a lot though, since you will be riding long distances.
Just for reference, the less touristy colored-leaves spots are Itamuro and Shikanoyu. If you ever plan to reconsider your route, keep these in mind too. Especially, Shikanoyu which is the oldest onsen in the area is situated in the old touristical part of Nasu, with Sessho-iwa, the shrine of Onsen-jinja, and a tourist information booth. My favorite spot is the old-fashioned cafe inside a shop called Michinoku (Yumoto branch). Through the big old window, you can see the side of a moutain full of autumn trees. The area, although authentic, can be a very low-key spot and boring for younger people though.
As far as traffic jams are concerned, try to avoid Nasu-Kaido as much as possible.