The Baltic Fleet could still reach the Atlantic
There are a series of rivers, lakes, and canals completely inside Russia that connects the Baltic Sea to the Black Sea:
The Volga-Baltic Waterway connects St. Petersburg to Lake Lagoda, then to Lake Onega, then Lake Beloye, and finally to the Rybinsk Reservoir on the Volga River. It's 369 km long.
Then take the Volga River from Rybinsk to Volgograd. I don't have the exact distance (web searches want to go by plane or car, not boat), but the whole Volga is 3531 km long, and on a map this segment appears to be at least 75% of that, so let's say 2600 km.
The Volga-Don Canal is 101 km long, outside of Volgograd.
Following the Don River, the distance from Volgograd to the Azov Sea is about 420 km.
Though the width and depth of this route is not as large as a sea route, there are numerous examples of Russia sending ships from the Baltic Sea to the Black Sea. The Minsk, Kaliningrad, and Korolev are Baltic Fleet landing ships that traveled this route and are now participating in the war in Ukraine. Wikipedia also reports
Analysis in May 2022 suggested that it may be feasible for the Russian Navy even to move its Kilo-class submarines between the Black Sea and the Baltic via the internal waterways.
Once in the Azov Sea, Russian ships can then navigate via the Black Sea, the Turkish Straits, and the Mediterranean to the Atlantic Ocean. This includes passage through Turkish waters, which is governed by the Montreux Convention. It's a loooong detour, but that's not NATO's problem.
That solves the issue of getting warships into or out of Saint Petersburg. However, Kaliningrad would be surrounded by NATO countries.