This issue is being investigated currently but the workaround from Skyrimguy is potential troubleshooting before a potential future fix. Other users also posted the following workarounds which may be of use:
"As soon as i spawned back in Dutch's bunker I...
-Opened the map
-Fast Traveled to Silver Lake Boat House
-Credits rolled again
-Skipped credits
-loaded back into the game.
-Fast traveled to the helipad at the Wolf's Den"
"1: As soon as you spawn in open your map
2: Air drop from a random location, (I don’t think it matters)
3: As your falling it should roll credits again
4: When you get back to the title screen it should be all spooky looking
5: Press continue again and you should spawn on top of the tower you climbed in the beginning of the game"
This happened to me as well, and in another thread on this forum the fix is that you need to click and hold the continue game for a couple of seconds, like we did when opening doors or such - remember we held the button for a couple of seconds. Try that and you will get to continue your game even after beating it. I have and have been playing further since doing that on my PC.