Hi,
Thank you for the details, it helps a lot!
If the live search is quick, that is a very good sign, that means there is only some sort of a conflict on the results page – probably another plugin.
Do you have any other search plugin active which may affect the results page? (SearchWP, Ivory, Relevanssi etc..)
If so, the please deactivate it and try again. It is possible that both plugins try to override the same query and it gets stuck.