The website of the British embassy where you are applying will provide information on the standard processing times for visa applications.
Remember that they are standards that an Entry Clearance Officers make an effort to meet, not deadlines that they are obliged to keep. Thus don’t buy any flight ticket in advance, as British Missions do not accept responsibility for any financial loss and do not fast track applications solely due to planned date of travel.
Premium Service Centres
If you are applying from within the UK for an extension of your visa or for settlement (“indefinite leave to remain”), you can book an appointment at a Premium Service Centre as part of your online application to have someone your application processed on the same day, unless extra checks are needed.
The Premium Service costs £590 for each person applying on top of the normal application fees.