I did this early last year to get the DVLA records corrected as part of which an age-related number is issued - my 1977 bike was registered in 1981, year of manufacture also recorded as 1981.
The covering letter I sent was dated 6th Feb, their reply letter was dated 11th March, so a bit over a month in my case.
I don't know whether that is a typical timescale but it is an example.
With Christmas / New Year thrown in to the equation I suspect it may take a bit longer.
Not sure whether you could contact them to enquire about progress.