Does anyone know how to clean the white stitching on leather straps?

I know you've already sorted your strap, which is great, but just for future reference if you clean it with soap and water and you happen to get your strap too wet when cleaning it, it's not the end of the world as long as you let it air dry. Don't put it on a radiator or anything to force dry it, just let it dry in it's own time.

Leather care cream / conditioner is fine to use, although you should probably restrict it to the outer surface or else you will lose the slight nubuck feel on the inner leather lining. However, when a strap gets older, the nubuck feel will go anyway, so at that point, cleaning it is probably more important than trying to retain the nubuck feel..... in which case leather care / conditioner is fine on both surfaces as it will replenish the natural oils.