At Battle of Hardhome, the slain immediately turned into wights. Yet other dead people, e.g. Jon Snow, didn't become a wight the moment he died1 and had time to be carried indoors.
Other characters at Castle Black talked about burning dead bodies, even when they were on the south side of the Wall2 at the time, as they were inside Castle Black IIRC.
1. This happened perhaps because he was south of the Wall?
2. Castle Black is barely south of the Wall