As previously being a Firearms Licensing Clerk, I wouldn't worry about speeding convictions they have no relevance to you owning a shotgun but still declare them anyway. If it was drink driving you might have to wait longer for your renewal/grant as they have to make enquiries to make sure your not a habitual drunk.
You dont need a reason to own a shotgun, it is only firearms (rifles) that you need a valid reason i.e Target. But if you apply make sure you have a good countersignature such as an accountant or postmaster. They will not accept Managers, Managing Directors or Company Secretary anymore.
Another biggy is if you declare Depression, it can take forever to get a letter from your GP or Psychiatrist and then even after a suitable reply from your GP it has to go to Supervision for a write-up.