Going through the composer interview from the SONIC CHANNEL sound column, these are the areas the present composers were asigned to:
Kenichi Tokoi - songs recorded in Rio like Main Theme (re-arranged for Title, Staff Roll, Results, etc.), menus, Rhythmic Gymnastics, Copacabana Beach (maybe a specific use that I don't understand? The loading themes, maybe?). Also sings chorus in Fukuyama's Copacabana Beach.
Mitsuharu Fukuyama - Copacabana Beach, original song Vento Brasileiro for Rhythmic Gymnastics, Rugby Sevens (and 4 other events). Also plays trumpet in Tokoi's Main Theme.
Hidekuni Horita - Table Tennis (both 3DS and Wii U versions), Triple Jump, Javelin Throw (and 4 other events)
Jun Senoue - BMX
Tadashi Kinukawa - Beach Volleyball
Tomoya Ohtani - 100m, Football, Sticks's Theme
Teruhiko Nakagawa -
3DS version: Story Mode Main Theme, Game Clear, various arrangements of Tokoi's Main Theme, Ground Theme arrangement for Rhythmic Gynmastics. Also he mentions Boxing as his favourite example of songs changing mid-game, so that might be his composition considering no else claimed it for themself.
Jun Senoue's mentions a comparison with the Moguls theme from Sochi 2014, so maybe he worked on it.
Tadashi Kinukawa's section also mention music from the two Winter Olympics games he composed:
Vancouver 2010 - Snowboard Cross, Half-Pipe, Skeleton (a total of about 20 songs, though I'm not sure if this is for that game or the series overall, a fluent speaker should take a look at that)
Sochi 2014 - Bobsleigh Skeleton, 4-man Bobsleigh, Snowboard Slopestyle, Ski Jump Large Hill