For the others who are going to have this scan (I know there's several of us) here's what my dsis says who used to do this job as a junior doctor - it might help to know what they are SUPPOSED to do:
That is an appalling example of what shouldn't happen ever.
Here is what we used to do (in Lancaster 7 years ago). The children would come to the child-friendly day ward with specially trained nurses and play assistants. They would have magic cream put on. They would have been told to expect to be there at least until after lunch but had access to lovely play room and helpers while there.
Trained paed nurses would explain to the parent how to hold the child in a tight cuddle, the nurse would then hold their wrist. The Dr (eg me) would then do the cannula. 90% of children didn't even feel it at all because magic cream does work but sometimes needs more than 1/2 hour. As junior doctors we had strict instructions NEVER to have more than 2 attempts at inserting a canula (same for taking blood) and would get a more senior Doctor to do it. The nurses saw it as their duty to keep the child calm and would have stopped the attempt if the child became distressed.
I didn't used to go with them down to the scan department but they all said it was fine when they got back. I'm not sure why the child was swaddled or whether this was usually done.
If you are worried I could go with you (when is it?). But you would just need to be assertive ie if X can still feel despite the magic cream, make them wait (you can still feel pressure but not pain), you can test this with a gentle pin prick with her looking the other way. Don't let anyone have more than 2 attempts. Stop if X becomes distressed and take as much time as it needs to calm her down. Ignore any comments about her scan being due now. Also you have every right to ask the doctor or nurse how long they have been doing canulas in children and if they are only just trained to do it, ask for a senior person to supervise them.