Natasha went to live in England with her children, sending George to boarding school until he died. After George died, Natasha eventually moved to France, where she died in 1952. Â
Neither Natasha nor George had a title of a sovereign (prince, princess, Grand Duke, highness, etc.) when Michael was alive or even some time after his death, but were known as Countess and when George was in school, Count Brassow. Only in 1928 did Natasha receive the status of Princess and George, Prince.
Interestingly, although the Dowager Empress hated Natasha and I'm sure didn't recognize George, what was left of her fortune/property was divided equally between Xenia, Olga, and George. So yes, George did inherit something.
Some of the Russian emigres mentioned George's name as the "true successor" to the throne and dismissed Kyrill, but George dismissed these claims. Legally, due to the circumstances of his birth (Natasha not being of royal blood) he probably would not have been eligible. Â
If you want to know more about George, Natasha, and Michael, you should read "Michael and Natasha". You'll learn all you want to know about them.