As for your example, both じょうげ and うえした will work and I don't think I can choose one.
The first reading じょうげ is the most versatile one that spans across a number of related meanings as well as grammar:
- n. top and bottom, upper and lower, the former and the latter (volumes), inbound and outbound (traffic); (adverbially) above and below
- n. all from top to bottom, high and low
- n. the quality of up and down, verticality, height, tallness, hierarchy, rank; (adjectively) vertical, hierarchical
- v. move up and down, fluctuate, bounce; n. fluctuation, bouncing
うえした and かみしも each shares a small portion of the definition #1 (and #3) shown above. You can refer to Ups and downs: かみ・しも vs うえ・した for their difference, but simply put, うえした is used for something really "vertical", and かみしも specially for those customarily assigned "up" and "down" and not very "vertical".
- うえした: for physical position, dimension, direction and hierarchy
- かみしも: for river (up- and downstream), stage ("upper" (=left) and "lower" (=right)), kimono (top and bottom pieces), tanka etc.
But うえした has one unique sense not in じょうげ: "upside down" by its own (= 上下（じょうげ・うえした）逆). The similar usage can also be seen in native words 前後ろ, 裏表, 右左 etc.
しょうか is... never used anymore, except that since it is all-漢音 reading, it must (prescriptively) be used instead of じょうげ whenever we read Classical Chinese. For other meanings kindly consult the dictionary, because they are foreign to us too :)
Hence, as the superior and inferior, they are able to have an affection for each other. (James Legge)