In WW1 you could appeal against the draft to a local tribunal and if you didn't accept its ruling, to a County one. Around here and probably elsewhere, the proceedings of these tribunals were summarised in the local newspaper, including names and reasons. Employers could appeal to keep irreplaceable staff. Typical reasons were working in an essential occupation, running a business, medical disability, or supporting a sick or elderly relative, all more common than conscientious objectors. Many people just had their join-up date deferred for a few weeks or months to make arrangements like finding someone to mange the business, or to get another medical opinion.
So you could try a local newspaper, but be prepared for a long trawl if it hasn't been indexed or digitised.