You only have to disclose convictions when asked or if convictions are mentioned in assumptions or terms/conditions of cover. You never have to mention spent convictions.
If you receive a conditional discharge your conviction will become spent at the end of the time period described by the discharge ie 6 or 9 months. However, it can still show up on enhanced DBS checks, making applications to stay in the UK/Citizenship, travelling abroad and so please read above if this applies to you.
You do not have to disclose any convictions you get during a policy until renewal, unless there is an explicit condition in your policy. If you are with an insurer who is making it difficult or more expensive then the Unlock website, where this information is summarised from, has a list of insurers.