The chamber can be reamed without much problem by a gunsmith with the reamer, heck a person with a little mechanical ability, know how and the equipment can DIY it.
Personally, I wouldn't do it. The Speed Sixes in good condition can be hard to find so I'd want it to stay original. But it's your gun you can do anything you want with it.
That said, a short scenario: You get it reamed to .357 Mag. You have it loaded with .357 Mag. You have to use it in a bad situation and you injure or kill someone. It will be confiscated during the investigation and possibly/probably a trial. The overzealous DA finds out that it had it reamed to .357 Mag. He will twist and hammer that fact to the jury ad nauseum to "prove" you are a dangerous person and were looking for trouble and had it done so that it could be used for nefarious uses. Sounds wacky don't it - but guess what, it happens a lot more than people think, especially in cities with liberal DA's.