There are remarkably few countries which actually ask any questions about criminal convictions. Assuming that you hold a British (or other EU) passport you can travel visa-free throughout the whole of the EU. Most other countries popular with tourists either don't require a visa or don't ask about criminal convictions when you apply for one.
The principal exceptions are the USA (which shares its criminal intelligence system with Canada and, I believe, Mexico) and Australia (which, to the best of my knowledge, doesn't share information directly with any other countries).
However, as TonyV indicates, on the rare indications that travellers are required to disclose criminal convictions, it's a matter for the individual country to decide what to do with the information. In many countries though, immigration staff are actually specialist police officers, so information gathered for immigration purposes might well be added to a local database of criminal convictions.