Thought I’d give some guidance on this, because I found some good tips which might be helpful to others.
First, this applies only if you’re applying within the UK for Stage 2, since different countries have different processes.
Second, what you should know is that the longest part of completing Stage 2 is getting a biometrics appointment. Now, not all people need to get an appointment - after submitting your Stage 2 docs and paying all of the fees, UKVCAS should email you within 5 days to let you know if you can do this online (my American girlfriend could) or must do this physically in person (me as a South African, and my Canadian friend had to it in person). Personally they never emailed me, but I logged into the UKVCAS portal one day and it said I should book a physical appointment.
Third, fresh biometric appointments are released each night at midnight for 4 weeks away. If you can wait that long, then this is a good option for getting a free appointment. You can pay more for premium appointments, but even those are still some weeks away.
Fourth, cancellations are released each morning at 09:00. These are usually for that very day, or within a few days. This means is you are pressed for time, looking for an appointment just after 9am each day will give you the best shot at this.
Personally, I logged into UKVCAS at 09:25 one Saturday, got an appointment at 11:30 and was done by lunch. 5 days later I had my BRP in hand. Very quick, very satisfying.
Best of luck out there.
