If you request a copy of your 'police record' yourself (which anyone is entitled to do, as long as they fill in the correct paperwork and pay the standard �10 fee), the police are obliged to send it to you within 40 days. If external bodies (e.g. local authorities responsible for licensing taxi drivers) can access the same data, it would seem to be reasonable to expect the same time frame.
However, obtaining your 'police record' is not the same as a CRB check. The Criminal Records Bureau's website states: "The CRB aims to process 90% of Standard checks in 10 days and 90% of Enhanced checks in 4 weeks.".
Chris