"Would it be better handled in the enum itself as each different case would require a different type of conversion?"

Do not put the conversion in the enum. The enum have no knowledge of the height, weight or temperature. So it has no value to compute. It only knows what is measured.

"Would I have to define the value property in the data model as a UnitType as opposed to a Double?"

When you are doing any kind of measurement, you want to store it in the most precise unit. That means you store your units in the metric system. Don't use Double. It is inherently imprecise. Use Int. Store in centimeters instead of meters for height. Use grams instead of kilograms for weight and so on.

And use Apple's Measurement API ;)