Most dealers will give correct delivery times at the point of ordering. Because of the volatile situation currently including the earthquakes in Japan and Russian war its meaning that the delivery dates and build dates are constantly changing and that means that it is increasinly likely that it will increase and not reduce.
Bear in mind I have a VW Golf on order since Dec 2021 which has had build weeks 11, 23, 44, 39 that I know of and almost everything in between.
Additionally, because of the new car issues - they're dealing with multiple customers a day when it comes to used cars. Typically a decent used car will get 5-10 queries a week. In some BMW dealerships, they're getting 15-20 enquiries a day one one vehicle which encompasses test drives amongst financing enquiries etc. Theres no real lack of availability of used cars whatsoever - there is plenty of stock, its just that most people know that they aren't going to pay the prices theyre commanding right now.
I think 3-4 days for an email is fine - then call if needed. I would only really ask for an update on the vehicle once a month in reality.