I realize this is a pretty broad question but just looking at the heirs of the houses of Hapsburg, Hohenzollern, Bourbon etc it seems like theyre still pretty well off.
Did they just have to go and get regular jobs or something and managed to continue to be wealthy?
Ertuğrul Osman, the grandson of Sultan Abdülhamit II, recently died in 2009, having lived much of his life in a nice if not exactly lavish apartment in New York. Unlike the current head of the Osmanoğlu family he actually grew up in Dolmabahçe Palace in İstanbul and was ten when the monarchy was abolished, so in a way he was the "last Ottoman" (if we want to use a term like like). because of the policies of the Turkish Republic, against which he bore no grudge to the end of his life, no member of the Ottoman House was allowed back into Turkey until the 1970s, and he himself did not return until the 1990s (I can only imagine what going back to Dolmabahçe would be like). His reception in Turkey says a lot of thing about the complex view of the Turks themselves on their Ottoman past, which is unfortunately somewhat outside of the scope of this subreddit.
Otto von Habsburg (1912-2011) lived a comfortable life in Austria, having been a vocal critic of the Nazis and had to flee Europe during WW2. He didn't return to Austria until the 60's when he renounced his claim to Austria, a move he hated to do. After WW2 he was active participant in EU politics advocated for a united Europe. When he died in 2010 a massive 13-day mourning period began in many countries his family ruled.
Most of them do have regular jobs like Otto von Habsburg as said above.
Simeon Saxe-Coburg-Gotha, heir to the Bulgarian throne and who was king of Bulgaria for three years ( 1943-1946 ) joined the US military in the late 50s before coming back in Bulgaria in 1990 after a long exile. There he was even elected Prime Minister of Bulgaria for four years.
Louis Alphonse of Bourbon, one of the French claimants has worked in banks in the US, Venezuela or Spain. He had previously requested to join the French military but like for his father his request was denied ( a law in France prevented descendants of royal families to join the military, law which might not longer exist as I recall an Orléan, the other claimant family, was killed in service while in the French army ) thus he joined for some time the Spanish air force ( he has the French and Spanish nationality ).
In an interview I watched, someone asked Louis Alphonse of Bourbon the exact same question and he answered You know, we do not have secret treasures and coffers hidden somewhere so we work like everyone else.