After finishing Wild Card:Finishing touches nothing happens, no popup-window, nothing. The Courier is not reacting on keyboard.
I have tried to use console commands and start "No gods no masters" and it starts but I suppose it's working incorrectly cause nobody attacks and in general it just don't complete it's subtasks.

So what should I do to finish the game? It's very frustrating to be at the very end and cannot finish the game properly

Even starting on the one of older saves does not helps - the same result

Do you use any mods?

No, no mods are used.

Funny. Can you upload your save game to hotfile.com
Post the link here. I can try it out on my game.

Just saw your message and I tried the save game out. It seems to work fine on my copy.
You are using a mod (conelrad radio it looks like) and all the DLC's.

I would first try to verify your game cache. There might be some corruption.
game cache:
1) Open Steam
2) Right click on Fallout: New Vegas
3) Hit Properties
4) Click the "Local Files" tab
5) Click on "Verify Integrity of Game Cache"

Test the game.

If that didnt solve the issue try vanilla fallout. Make a new manual save first before doing anything.
1. Uncheck all .esm and .esp files using the launcher or FOMM if you're using that. Leave falloutnv.esm checked.
2. Now navigate to the data folder in the new vegas install directory (default: C:\Program Files (x86)\Steam\steamapps\common\fallout new vegas\Data).
3. You should be in the data directory. Make a new folder called nam files and MOVE (not copy) ALL the files with the extension .NAM into it (deadmoney.nam, oldworldblues.nam, etc). There should be a .NAM file associated with all of the DLC's you have. This step has to be done otherwise the DLC will still load.
4. Test the game. You will get a message saying some items will not be available since you have disabled the DLC's that give those items. Click yes to continue. You won't see your DLC items but dont be alarmed. As long as you dont overwrite the save you loaded then your items won't be lost. You also made a backup save (the new manual save) so don't worry. You won't have to save at all since this is just for troubleshooting purposes.

The game should make a automatic save. This is a system save. Then you should get the popup message asking if you are ready to go to the hoover dam.

So here's a couple things to try. Post your results and we can go from there.

I finally resolved the issue by reinstalling the game. Thanks for your help!