Can my doctor send my prescription to any pharmacy?

Yes, with few exceptions. Modern prescriptions are mostly electronic: your doctor's system sends the prescription directly to the pharmacy you name, any licensed pharmacy you choose, and the NHS equivalent says the same, that your GP surgery may be able to send your prescription to a pharmacy of your choice. Changing your mind is easy: if the prescription went to the wrong pharmacy or you would rather pick it up elsewhere, call the new pharmacy and they can usually pull it over, or ask the doctor's office to reroute it. The exceptions: controlled substances in the strictest federal category, Schedule II drugs like oxycodone or Adderall, have tighter rules, including no automatic refills, and some states restrict transfers of controlled-substance prescriptions between pharmacies, so those are harder to move once sent; and specialty drugs locked to a specific specialty pharmacy network cannot go to your corner pharmacy at all. The practical tips: save your preferred pharmacy in your doctor's record so it is the default, double-check the pharmacy name when the prescription is sent, especially after a move, and if a pharmacy says a prescription never arrived, the fix is usually the doctor re-sending it, which takes one phone call. You also never have to fill where the prescription first lands if you have not picked it up yet.