Airline lost my luggage at budapest airport?

I had a few baggage losses (delays) around Europe (and once in Brazil), and my current picture is, as long as you can file a proper 'lost baggage' claims, and have an address where they can deliver to, they will do it within a few days. It seems to me that around a few percent of the luggage doesn't make it aboard, so the delivery game is pretty standard procedure.
In Ferihegy (as other places) ground handling is done by a company other than the airline (here it is Celebi, s. blue logo on the runway-busses, http://www.celebi.hu/en/contacts.html ), so as soon as the luggage is on ground I think it does not to matter much which airline carried it.
It is important that you don't loose your case reference ID you got when you reported the loss!

A few weeks ago, my budget flight (London-Budapest) forgot my luggage there, it was delivered the next day or so. I had a similar experience last year on a Lufthansa flight Berlin-Vienna. So, I'd say, don't worry too much about it, unless it had... fresh flowers to your girlfriend straight from Brussels. The annoying thing is they rarely know the exact time they will turn up so you may have to wait at home.
After one day I'd still call just to check if they haven't forgotten about you.

But, there is still something to worry about - theft. Last time, I had some Hungarian change in the outer pockets of my bag - and that was gone. I wonder if they would really go as far as open it to steal something from the inside (I hope, only untrackable items, like cash, might be in danger). This is hard to counteract in this case, as they delivered the bag locked with plastic rings (against theft, I suppose...!), and I needed scissors to open my own bag - of course I did that after the delivery guy left.
So, I'd advise you to keep the scissors/knife at hand and open the baggage in the presence of the delivery guy (tell him to wait), to point out immediately, if something is missing. (I do hate how they steal from customers and would love to see a proper report filed on such a case!)

From the above it can be seen how important it is to get each piece of luggage clearly marked with your name, hopefully with ID. Once we simply wrapped a plastic bag around a rucksack and poster roll, following the advice of the check in counter clerk - the plastic bag opened and the poster roll was never seen again - now official ID, not even good markings - nothing possible (Air France, Berlin-Boston, 2006 - btw, avoid aifrance if possible:).