There are 2 issues here:
Firstly - the prescription and how the situation was handled. The prescription was signed off by a GP. Responsibility for a prescription lies with the GP and to some extent with the pharmacist who hands over he medication - both of whom are medically trained to know what drugs are being prescribed for what and to query where a drug or particular dose might not be right. A receptionist will not generally be qualified to question a prescription, however, if she genuinely was concerned that there might be an error, the CORRECT thing for her to do would have been to politely, quietly and discretely ask your DD to take a seat for a moment while she just quickly verified that it was indeed correct. She should have then just checked with the GP that it was right and then given it to your DD. Simple. A receptionist has no right to refuse to hand over a prescription that has been issued and approved by a GP.
A pharmacist might query a prescription to be on the safe side, and rightly so if they believe it to be wrong, but again this is between the pharmacist and the GP.
If there is no good reason to deny a patient a prescription, particularly a repeat prescription for a chronic condition, then refusing to give the prescription is potentially very dangerous territory - especially if the patient suffers adverse effects as a consequence of not having their medication.
The second issue is confidentiality. Doctor's receptionists MUST ensure they do not breach confidentiality and data protection. Loudly discussing a patient's condition/medication like that in front of an audience is a potential confidentiality breach and a serious training issue.
Your DD should definitely raise a complaint - not necessarily about the receptionist verifying that the prescription was correct, but about the way in which the whole thing was handled. Accusing your DD of double dosing was inappropriate, refusing to hand over a prescription was also inappropriate and the breach of confidentiality issue definitely needs to be flagged up to the practice manager. It is important that they are made aware of this.