I'm afraid it can be as long as the proverbial piece of string.
Part of the problem is that whilst memory clinics can usually decide whether someone has some sort of dementia, it can take a very long time to identify which condition a person has. Dementia can have many causes. Increasingly, as well, clinicians seem to come up with a 'mixed dementia' diagnosis as if covering their backs.
My wife was referred to a well-regarded memory clinic in 2000. She got the following diagnosis in 2011: Probable Alzheimer's disease (posterior cortical atrophy). Probable !! Since then she has had an additional diagnosis of Parkinsonism.
But during the many years of waiting we got on with life as best we could and created many happy memories. If you think a quick diagnosis will help then it might be a case of being careful what you wish for.
Now I know some people can benefit from some medications and that a diagnosis is usually required before drugs are presribed but I also know, from our experience, and from my reading that this is not always the case. S was tried on a number of meds before she had a diganosis.