I think you're worrying needlessly. Assuming your previous employer was unaware?
The job strictly speaking can't ask you to declare spent convictions, which yours is. If they discriminate against you it's illegal.
Meanwhile in the real world, have you got an unusual name? Have you googled yourself? Try a few different search engines. Is there a link, references or people linked to the employer who know about your past?
I usually advocate making a stand. However this job isn't exempt and your conviction is spent. You legally do not need to declare.
.....and they haven't asked for spent convictions (just as well because it would have been illegal). A bit unusual that they haven't asked for unspent convictions, was there a waiver saying the job was dependent on these checks, even then they are putting themselves on a sticky wicket. If they have offered you a job and then rescind this because of a spent conviction and formally document this is the reason, it's a law suit waiting to happen.
In a year or so, this conviction will be filtered completely out of the system, so unless you are working as a policeman or M15 you can apply in bliss, even for roles like a nurse or social worker.