When initially applying for a Personal Licence, 'spent' convictions do not have to be declared:
http://hub.unlock.org.uk/knowledgebase/pub-licensee/
Any conviction thereafter for a 'relevant' offence (which, for example, includes drink-driving but doesn't include careless or dangerous driving) has to be notified to the licensing authority.
As far as I can see from their website, the Financial Conduct Authority doesn't impose any duty upon any person (who has already been given FCA approval) to notify them of any criminal convictions which they might acquire after they've been granted approval but the courts may well pass such information on anyway. (As an analogy, teachers aren't legally obliged to tell the GTC about any convictions they acquire but the courts automatically pass on details of things like sexual offences).
To the best of my knowledge, a similar position applies to both of the Chartered Institutes for accountants.