The Models 10-19 are all the same K-frame and the only difference between the M&P models (10, 13) and the others is sights and barrel thicknesses. You should be able to screw that M14 barrel right into the M10 frame.
The only problems I can imagine is making sure the front sight is pointing straight up while you still have a cylinder to barrel gap of between .003 and .006", and using the M14 front sight with the M10 rear notch. You may need to find yourself a rear sight assembly for a M14, M15 or M19 and have the frame milled to accept it.
The extractors should be the same length on all of them except for the M13/19 which are both 357s while the others, the M10, M12 (I think), M14 and M15 are 38s.
If you aren't going to try and install rear sights to match up to the front sight and just want a new barrel you can always call or e-mail S&W and see what barrels they may have laying around that will work for you. I have found them very helpful. HTH.
