Based on the Seat 61 page on Hungary, one option from Budapest to Slovakia is probably train to Vienna, then onwards from there into Bratislava

As for everything dissappearing in 2 months time, that's to be expected. Early December is the launch of the next Europe train timetable period, and train bookings are only just opening beyond that horizon. (France was last week, as was Germany, not all open yet). I'd suggest trying again in a few weeks, once the rest of the January-June trains have been announced + loaded into the various booking systems