I think it depends a lot on where you live......but also on the urgency of the situation. If the person being looked after is vulnerable or at risk, you should make that clear, in writing if possible, as cases are often allocated in order of priority.
Same goes to some extent for Carers assessments.....though you can often complete these online.....again, depending on where you live
As everywhere, the more noise you make the better you get heard. The first, inconclusive assessment didn't take too long to get done but didn't help at all. So for the second I threw a total wobbler, called it an emergency, said the right words (vulnerable adult at risk) and even got the GP involved who called them as well and what do you know, I had a social worker on my doorstep the very next day. I told him exactly what I wanted, he made some discouraging hm hm noises then approved it all.
I went to see on duty psychiatrist today regarding my stress and anxiety,a lot of it caused by the anti social neighbours. Psychiatrist rang housing,they told her that they would ring me. Got no call. Im at my wits end.
I've had two assessments for my mum, in two different councils. Neither took too long to get done but what did take the time was then getting the proposed care plan signed off. I had to make formal complaint in first instance to head of social services and within 24 hrs care plan signed off and auctioned!! From assessment to plan in action took 3 months. 2nd council slightly quicker but I phoned at least once a week chasing all the time. Very stressful and not what you need when you have enough on your plate with a parent with dementia. Sadly you have to keep badgering.