There are various online pharmacies that will dispense to you if you have a valid UK prescription.
I aked a family member who can legally write prescriptions (how vague is that ) what would pass for a private prescription and she confirms to me that in extremis so long as it is not a VERY controlled drug (there are different degrees of control, but from my understanding dihydrocodeine would not be a controlled drug for these purposes despite being a class C (or B?) drug under the misuse of drugs act - I think it basically applies to phenobarb and the stronger opioids but am not 100% sure Edit: I have double checked and the degree of control is avaliable for you on the BNF, as described below - just look up the symbol on the info page for the particular pharmaceutical. Those guidelines are at BNF site - (free signup required) all it needs is a signature and name of the prescriber and unambiguous details of what the drug is (including dosage etc).
Of course if a prescription was written on a discarded napkin it is likely that the pharmacy is going to make further inquiries.
Finally, I note that anyone may register to the British National Formulary website for free. If you do so, then take a look here you can get very detailed notes on how to write a valid prescription, if you were permitted to do so - bearing in mind my previous comments about how a private prescription need not necessarily confrom to any particular layout.
I do not suggest that anyone fraudulently writes a prescription. But the above information is interesting to note in my view, and perhaps would be of help if anyone wanted to.