"Why is Gypsy music not Hungarian, when does an immigrant music become part of the culture?".
   Good question, and one which can be political dynamite. Arguing on this topic has cost many people their jobs, their friends and their lives, if you take a look round various nationalist political hotspots of the past couple of centuries.
   I've made no study of Hungarian gypsy music, so I won't try to answer directly. But from what I have looked at, my opinion is that for immigrant music to be considered part of the culture of the new(host) country (as in "traditional culture"), it's got to have changed. Not assimilated necessarily, or even fused particularly, but it must have changed. So cajun music, for example, is unquestionably Louisiana American music, even though it remains distinct, and is sung in French. It is obviously American, because there's nothing like it in France. But recently arrived ethnic immigrant music, sung to bolster identity or whatever, doesnt seem to qualify as part of the traditional culture of the new country till it's bedded in and started to mould itself a little. Flying Morris Dancers to the South Pole wouldnt make it Antarctic Dancing. They'd have to live there a while, change the steps, modify costumes etc, before they qualified.