IMHO, Jersey has too much fat for hard cheeses. You can get away with it if you eat the cheeses when they're young and creamy, like for butterkase. Otherwise, you tend to get somewhat odd flavor development, and the curds behave oddly (retain water, are too soft, etc. I would skim and use the fat in bloomy rind cheeses like triple cremes, or for butter/cream cheese/mascarpone.
You can get away with straight Jersey milk for cheddar during mid lactation, if the cows are on pasture only. That tends to be a better PF.