Welcome to this user forum.
The fact that there is a line already, does not help, unless its still a working line and the previous owner has not yet cancelled it.
If that was the case, then a Working Line Takeover can be slightly quicker, as this would enable the phone service.
If the previous owner has cancelled the service, then all the cancellations tasks have to be completed first, before your phone service can be activated.
At that point, you will be able to see what broadband options are available.
Unless you asked for a later install date the answer is no, you're automatically given the next appointment.
10-14 days is the typical lead time for provisions and upgrades.
I was asked if I wanted to take over the existing line when registering and said yes but then the first available dates were still after 15 days. Is this expected?
Yes, that would be about right, allowing for the minumum 10 day delay that Ofcom require, to reduce the incident of "slamming"