LRI or Kampfeld would be places to call. Anyone with a lathe can recontour a barrel, but it's not as common a practice as it used to be. Price will depend on what you want done. A complete guess is about $250 thinking about how long it takes me to do one and what I'd charge for shop rate.
Sounds like, 9/16" is the right thread size for a tikka sporter barrel cut down to 20"? And then you could put an adapter to get to 5/8". BUT, the nomad ti has a screw in insert and you can get it in 9/16-24. Could you just use that and direct thread on? Or is straight 9/16" not enough...
I dont know what all goes into doing these things for a smith but it seems like it should be easier than some of the elaborate fluting jobs but maybe more metal removal? I do have one barrel that I might send in to be recontoured if it doesn't sell..