Not sure about the criminal records provider, but the term "Background check" is a broad term which includes criminal checks, working with children checks etc.
I don't know who the largest provider of National Police Checks is but I've seen police checks from Precise Background Services and InterCheck in the past.
Crimtrac are responsible for the accreditation and management of National Police Check brokers under the National Police Checking Service and you may be able to contact them for an answer to your question.