(2-part post):
As Norman states, in his usual forthright way, the immigration authorities of foreign countries do not have direct access to criminal records. The UK authorities do pass on certain specific information, relating to certain offences, but simple possession of illegal drugs is unlikely to be included.
You'll definitely have no problems entering any EU member state because you have the automatic right to travel to (as well as to work in and live in) any EU country. (That, of course assumes that you hold a British, or other EU, passport).
Many other countries don't require UK passport holders to hold visas when visiting as tourists so, once again, you won't have any problems entering those countries.
Most countries which require you to obtain a tourist visa don't ask any questions about criminal convictions so, yet again, you'll have no problems.
The USA requires anyone who's ever been arrested (even if totally innocent of any crime) to obtain a visa prior to travel. It's a lengthy process, taking up to 5 or 6 months. If you apply for a visa, it's likely that you'd be refused. However, as is frequently pointed out here on AB, countless people enter the USA under the Visa Waiver Program by simply 'forgetting' about their convictions and answering 'No' to the relevant question on the form which travellers complete on the plane.