Hello,
I have never been to this area personally. But have looked up some information.
For Byakue Dai-Kannon, you can take the Gururin bus from Takasaki-station (15 minute train ride from Karuizawa). Then it is a 10 minute from the bus stop. According to a Japanese blog, the road to the statue is smooth, but there are only stairs at the bottom of the statue. (http://www.odekake-travelhelper.jp/thblog/234683). I sent an email asking about accessibility.
To get to Lake Haruna, you would take the train to Takasaki Station. But after that the only way to get to the lake is by bus ( but cannot find any info on accessibility of the buses, some images look accessible, but cannot guarantee it)…
To get to Akagi Nature Park, you must take a taxi or shuttle bus from Shibukawa station. I could not find information, but from a picture, it looks like the shuttle bus is not accessible. Also, a different wheelchair user’s blog says that only the area near the entrance is accessible (https://guide.ikiru-chikara.org/akagisizenen/).
It may be best to rent an accessible taxi: http://koha-fukushitaxi.com/company.html
I hope this helps