Of all the cars I've bought in the last two years (which is a few) none took longer than 4 working days. I think the DVLA are shite hot at changing V5c details TBH. Bought my lastest on Friday 2nd June, got the new V5 on Wednesday 7th.
You said the last owner waited until the end of the month before contacting the DVLA, so you could save a month's tax. Are you sure they actually did let them know? :? Or maybe they sent the old one by post rather than online? Or maybe the previous owner got your address wrong?