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Going back to your query, in general, there is no legal obligation on employers to provide a reference for ex-employees, unless it was a contractual obligation for them to do so or it is for a specific financial services jobs which perform ‘controlled functions’. It will also be discriminatory if a reference is refused because of a protected characteristic, which includes age, gender, race, religion, disability, sexual orientation, pregnancy.
Therefore, unless the refusal was due to a discriminatory reason or you worked in one of these excepted roles, which you don’t, there would have been no legal obligation on the ex-employer to issue you with a reference and even if you lost a job as a result, you will not be able to challenge them legally.