As someone who regularly rides all three, it really depends on the rider, the equipment, conditions, and how you ride. People lump 'bents together, but there's more difference between them than there are uprights even if you consider ridiculous differences such as BMX and race bikes.

A trike will likely be noticeably slower everywhere due to heavy weight and large frontal area. A working number might be 2 mph slower than an upright which will be a little slower than a recumbent on flats but faster on climbs.

Note that some aerodynamic recumbents can make your neck and back feel far worse than your upright bike will. In all honesty, my race bike is the most comfortable of the lot because it's much easier to change position and rotate muscle sets than on a recumbent or trike.

My gut reaction for a tour like you're planning would be to get a LWB recumbent. Touring is what those bikes do best, it will stable and you'll have a great seating position and view. You'll get good speed on the flats but you'll also probably be slow on the climbs.