First, wash your strap with soft water and bar of soap, then let the strap dry, when the strap is dried, use a vegetable leather conditioner, ensure that the leather absorbs it (meaning apply it little by little and slowly), let dry, rinse the strap with soft water, and apply your conditioner again, best results are attained when you have 3-4 layers of the leather conditioner.

Now, do not laugh at me, when you are done with the above and when the strap is completely dried, apply a tiny little of neutral ‘cire d'abeille' beeswax, this will fix the conditioner, let dry...
imo you will end up with one of the best the best old new leather strap... beside, beeswax will help you to protect your strap against splashes...

Or do the dishes... J