The last one for my household was about a month.
That may not be usual, but it probably won't be much quicker.
All the times add up. You have to be able to book and complete the course, for example, well before 6 months after the offence, because otherwise the prosecution can't be brought against you.
But if you're reasonably prompt, as I think you were, and you're the registered keeper so you're not already part way down a chain of referrals, then it shouldn't be an issue, the process will cope.
For us:
- 8/9/22 offence
- 15/9/22 letter sent off
- 12/10/22 offer of course etc.
- 19/12/22 course booked and taken