So I tried to load and unload transporters (APC aka "BMT" type military transporters).

Tried to put some Toughs (bazooka robots) into an APC but nothing happened.

Tried to move both units to the same location but nothing happened.

Is this a bug or do you need to press a certain key to put troops into transporters?

remake have no keys at all. just mouse ("touchscreen") control. Although first old PC-Win port of original DOS version has plenty of hotkeys.
as i remember PC-Win version, there wasn't any button keys related to APC or anyone (except [G] for guns). As I remember remake i has no issues with APC. There was legacy manuals, i would first check them, Steam Community hub and Internet Archive. Sony PS/Sega SS manuals are good too, i doubt APC there work differently.