Currently, here in Australia an enema (fleet, MicroLax, barium) can only prescribed by a doctor since they are regarded as a medical treatment and/or medication. I know this because I work in a hospital as a nurse assistant.
Suppositories which are mainly used to relieve constipation must also be prescribed by a doctor since they are also regarded as a medical treatment and/or medication, because many of them also contain pain killer drugs like paracetamol in addition. Worth noting that some drugs do come in the form of suppositories and are used in situations where a patient (for several reasons) may not be able to take medication orally or via IV injection.
However, with regards to your question 'Hospital Enemas On Whose Orders?', I think the answer will depend in what country the hospital/clinic is located, and what time decade period.
Perhaps currently in some countries nurses are legally authorized to administer enemas without a doctors prescription. Perhaps this was the case during WW1 and WW2 in military hospitals where doctors were few and nurses may have had the authority and necessity to make clinical decisions/judgement based on their previous training and experience. (Feel free to correct me regarding this point).
As I have stated in some of my other posts, as a child I was admitted 8 times in a hospital in Rome between 1972-1976. Back then and for that part of the world, it was the hospital policy to administer pre-op enemas to young boys. I am not able to speak for for any other sex or age group because I never witnessed anyone else besides myself and the other boys receiving enemas.
Because I never dared to ask any questions, I am not able to answer if those enemas were individually prescribed by a doctor for each individual boy, or if it was the standard hospital practice policy to administer pre-op enemas to boys/girls/teens/adults. If it was the standard practice policy to do so, then it is possible that those nurses may have had the authority to administer enemas without a doctor's prescription for each and every individual patient on the operating list.