With great difficulty. You may need to use the Keying>Extract effect to try to pull a luma key. And then use a pretty heavy color grade to make it a color.
The problem is, you can't "change to color" because a white shirt has no color to change!
Do you have an example image we could see to offer some better, more specific advice?
The degree of difficulty depends entirely upon the shot. If it's what I suspect it is you can duplicate your keyed layer, pre-compose the duplicate and then rotoscope by hand or use Roto brush to isolate the shirt. If the shot is more than a few seconds long you will probably want to render a good digital intermediate before you Rotobrush or rotoscope. I use this technique all the time to fix small problem areas in shots I am trying to key. The keying usually does most of the work cutting out the shirt so it should not take you too long to do the Roto because you only have to worry about the collar and cuffs or maybe if I hand moves in front of the shirt, you will need to get rid of the hand.
Once the shirt is isolated you can use it as an alpha track matte for a solid that is the color of your shirt with the blend mode sit to multiply (Which may be the easiest solution) or for an adjustment layer with some color effects applied. Colorama may be a good place to start. It all depends on the shot so a screenshot and an accurate dcription would really help us point you in the right direction.
It is a lot easier to see your screenshots if you just drag them to the reply field on this forum. They can be full screen and it will work great.
Go into your pre-comp and copy the footage or duplicate it then precomposed the duplicate like I explained so you have a comp with only your actor and no background.
You can then use Roto brush to isolate the white coat and if you wanted to you could manually run a scope away the white T-shirt so you're only changing the color of the long sleeved shirt. Pending on how much he moves his hands you might be better off just doing everything by hand. The nice thing is you don't have to worry at all about 90% of the shirt because the keying already took care of those edges.
Aside from what the others already said, the issue with white stuff always is that it looks completely fake when coloring, even more so since your video was shot with too much exposure. If at all it should have been shot with one or two stops below the actual value, then the colors completely be recovered in post. In the end, it may just be simpler to re-shoot this.