Seems like a ridiculous hard to believe situation, but you are perfectly entitled to have your car repaired wherever you choose.
Why are you? Never seen a FCA ruling that a policy terms cannot restrict who covers repairs as long as its adequately called out as a deviation from the norm prior to purchase and in the terms sent out post purchase. Haven't heard anyone tried to do it either though.
To the OP - I am confused who you are actually talking to, and fear the worst. In most circumstances of brokered business you'll simply have an insurer/underwriter and a broker but you seem to suggest there is a third organization involved? Are they a claims management company? Was this a "non fault accident"?
If it is the underwriter that's managing the claim then its not unusual for things to go slower if you choose not to use their approved repairer. Approved repairers often have delegated authority to commence repairs as soon as the cars in as long as there's no suspicious damage etc and its clearly not beyond economical repair. Where the work goes beyond their authority they often have imaging etc that goes directly to the office based engineers that will authorize it within a day or so.
Non-approved repairers often require an engineer to physically go out and visit a garage, make sure the repair quoted matches the damage done, negotiate prices etc. As insurers have invested more in their networks they've reduced the headcount of engineers and so a chap may only be in an area once a week/fortnight etc
If its an accident management company and a non-fault accident then all the above goes out the window. They are doing everything on credit with a view of adding a big margin and then claiming from the at fault party's insurance. As such they often don't like actually extending the credit for the repairs until they are comfortable that the third party was insurers, no indemnity or liability issues etc.
