Well, they would do what Heinrach will say, but note: They have been preparing a large expedition into the Rtassian Valley, so even with the change in leadership they can't afford to change too much of their plan of "ally with large horde, turn them against legitimate enemies, protect allies" without their large horde turning against them.
Right now they are allied specifically with Danyaria and Xilexis, both of them have large amounts of Iron Faith adherents, which is a rather strict form of the Grey Faith with Iron Prophet teachings (which I think will fade as time goes on). Danyaria also has a surprisingly large amounts of forsworn.
As for other Buggerites, you can't really finish conquering them as they range from OTL Poland across the Danube all the way to the Balkans, but they have mostly stopped migrating by now.