I've been in this situation a few times with Qatar (and twice with LH). You file the report with the carrier that you arrive on regardless of who else you were traveling with and it is Qatar's responsibility to pay compensation or take a claim in this case. Your PIR serves as a receipt for the bag tag. Qatar has told me in the past that they will pay compensation to replace items if the bags are not delivered within 24 hours of your arrival. They do not like to pay or pay very much so you may have to insist.

LH in my experience is excellent with tracing and rerouting bags. If your bags are with LH in FRA they will most likely check on your location and route them. Since they have I think daily FRA-JED flights I would wager they will put it on their flight. Their baggage handlers in FRA could decide to reroute the bags if that makes sense and in that case they wrap a new baggage tag over the original one with the changed routing. There is a preference for a direct flight and for the original routing, so both argue for using LH FRA-JED.

I've found UA's 1K baggage service in the USA very helpful as a backup and with much more information than Qatar would give me. Qatar's attitude has been that when the bags show up we'll let you know, generally I find they get the bags in the property office about 2 hours after the flight lands. The good news is that they generally do show up. With the snow in Europe maybe it will take an extra day or two.

Saturday's FRA-JED flight arrived quite late. Is it possible the bags were on that flight and you'll get them today?

You can view the PIR status on the web and expect it to be updated around 2 hours after the relevant flight lands. In my experience QR does not update the status until they have the bag in hand in the property office. Other airlines might actually update the status while the bag is en route.

http://www.worldtracer.aero/filedsp/qr.htm

As I said I found UA's baggage service had access to a lot of useful information in the system (actual information about bags loaded onto a flight).

It's good that you are staying the week in JED. If the bags don't show up by the time you leave JED, file a claim for lost luggage with QR. But in general odds are excellent that you will get the bags back at some point.

Not sure if you did this, but any time you change airlines internationally it is a good idea to show the actual bag tags to the transfer or check-in agent. In some cases they will enter information into the system that helps the bags get on your flight. I'm not totally clear on this, but many agents have told me it helps.