It should be possible, but I personally would try to handle the luggage differently. The taxi ride back to Nakatsugawa sounds time-consuming and costly.How about the following plan?
Leave your luggage at Matsumoto Station.
07:15 depart Matsumoto by limited express train
08:08 arrive Nagiso
08:15 depart nagiso Station by bus
08:22 arrive Tsumago
WALK TO MAGOME
11:55 depart Magome by bus
12:20 arrive Nakatsugawa Station
12:49 depart Nakatsugawa Station by limited express train
14:03 arrive Matsumoto
Get you luggage
14:26 depart Matsumoto by local train
15:40 arrive in Nagano
15:58 depart Nagano by Nagano Dentetsu
16:49 arrive Yudanaka Station
3.5 hours should be ample of time for the walk (good walkers should be able to do it in around 2 hours), but you might not have much time to see much of the two towns themselves. Note also that museums in Tsumago do not open until 9am.
Note also that Shibu Onsen is 2 kilometers from Yudanaka Station.
