You can find some of it in Washington DC at the Hillwood estate. Marjorie Post, the heiress of the Post food tycoon happened to be married to the ambassador to Russia during the infamous 1937. She had a good eye for valuables and swept through what was left by that time from the tzar's treasures, purchasing things in bulk. It was lucky she was there because she was probably the only one around who cared and was alive at the same time, given that it was at the height of Stalin's repressions aimed at the most cultured and educated.
Thus she ended up saving the actual crown in which tzarina Alexandra was crowned. She also owns the china set used by Catherine the Great to personally treat the war heroes to an annual banquet. Marojorie Post's collection is priceless but she was a sheer serendipity. The thought of what was destroyed through ignorance and willful "cleansing" is quite depressing.
Some Romanovs' possessions remained in Russian museums. There were always people who would give their lives to protect art and many did. Sadly, according to some accounts, one of the mayors of St. Petersburg, then Leningrad, had the last name Romanov and he used Romanov's named china for his daughter's wedding, breaking some for good luck.