Both Australia and New Zealand use an interactive Advance Passenger Information system known as Advance Passenger Processing (APP). Under APP, check-in staff are required to verify a passenger's immigration status directly with government systems, which will then respond with a board/do not board message.
The contact numbers exist for check-in staff to call to resolve any issues with the APP process. For example, a passenger may check in using a different (newer) passport from the one they specified when they applied for a visa. The contact centre would be able to transfer the visa across to the new passport number so that the APP process can be completed successfully. A member of the public would not be able to get assistance from calling these numbers as they do not have access to the APP system.
As far as I know, the UK does not use interactive Advance Passenger Information systems, and relies on airlines performing manual checks. Therefore it would make sense for them not to have (or not to have publicised) contact numbers for clearance queries.