I used to be involved in contracts for mammography for a UK county. Some is done on fixed sites, but a lot is delivered using mobile units parked in towns so patients don't have to travel far.
The machines are all more or less the same and have a kind of flat surface you lean against with one arm raised while two acrylic plates first squeeze the breast horizontally and then vertically, which involves a change of position. Four shots are taken (horizontal and vertical for each breast) and while it isn't comfortable, it isn't so unpleasant I've heard of anyone not returning because it hurt too much.
For fit women I've not heard of a yoke being used and having often been in the mobile units because every couple of years there was one in our car park, I've never seen anything resembling one either. Even for patients who've lost the use of an arm, there's usually a second radiographer around to hold it out of the way, so I can't quite imagine why Kat had the experience she did. Weird!
Apart from dense breasts, the major issue is imaging the tail of the breast, which extends toward the armpit and dealing with smaller breasts, which are more difficult to trap between the plates. The fall back investigations are ultrasound and MRI but like X ray, both have their advantages and disadvantages.
The UK breast screening service makes provision for trans people and even has protocols for non-binary!
There's a separate protocol for people with augments because of implant rupture risk and the TL;DR is the least possible pressure between the acrylic plates is used. There's no evidence a mammogram has ever ruptured an implant, but a mammogram isn't a valid way of checking implant integrity either, so the advice is that if a rupture is suspected, a mammogram shouldn't be done. If a rupture is known to exist pre mammography there's an entire protocol called the Eklund technique. It's all pretty well sorted, considering.
