You can do this with a 12 volt wall wart.   I have done it with a bench power supply and I have seen kits to convert the Cold Cathode over to a wall wart.  

One thing to think about is the heat that comes out of a cold cathode.   They generate a fair amount of heat.   LED's generate less heat for the same voltage and light output.  

I know that a cold cathode will output a smoother light output with no dark spots.  But if you work at some of the LED tapes that are out there, you can them dark spot free at the same voltage, and lower heat output.

If you do go with the cold cathode, I would monitor the heat output when you first put the model together to see how long you can leave it on before it gets too hot.