Due to the gun not originally having an underlug the profile of the front of the frame most likely won't meet up with the Model 19 underlug. Also being a magnum caliber the Model 19 cylinder may be longer. I'm thinking that sticking with Model 10 or 15 parts would make life a lot easier.
The model 13 was a .357 based on the model 10-6. It was in production for about 20 years with the last year being 1999. I measured my model 19 cylinder and it is indeed longer than my model 10-6. I suspect that the difference in length is compensated by the shorter forcing cone of the 19. It should be noted that a model 19 barrel might be rather difficult to find. If you want to make this conversion, it might be easier to find a model 13 cylinder and barrel.
Possible? Yes. Practical no. The cylinder would have to be shortened & timed, some lathe work need to be done to the barrel, etc.. It would be much cheaper to buy a Model 19. A replacement Model 10 barrel would need some work but much cheaper if it truly needs replacing. There are no drop-in parts except grips.