When did drugs become illegal in the United Kingdom?
I assume you mean the more commonly used illegal drugs such as heroin and marijuana, but the variety of so-called "controlled substances" has been amended from time to time to include a whole list of different drugs, legal and not.
But overall, most of the controlled substances were made illegal, at least without a prescription, in 1971 with the Misuse of Drugs Act. While some may have individually been criminalized before that, this was the most sweeping set of laws ever passed in Britain on the topic of drugs, and is still the cornerstone of British drug policy.
The link below includes a site with a comprehensive list of scheduled drugs and the year in which they were made illegal.