I have replaced the leather on a few, drill out the rivets from the bottom. You will need a flat surface to lay the seat on to peen the new rivets. A flat piece of steel or an anvil works best. The easy way is get someone to hold the saddle so the rivet you are peening is flat against the leather. Then using a small ballpeen hammer [a piece of round stock steel will work also with a larger hammer] peen the rivet so it is tight against the leather but not so tight it deforms it. You have to do the nose first and the side ones are tough but just take your time. When you get the rivets tight you can then peen the top of the rivets so they blend smoothly with the seat. A small tack hammer works well for this, be careful not to hit the leather as it will dent.
Good Luck!

Be careful when drilling out the old rivets so they don't spin and damage the leather. You could use a small
grinder instead, like a dremel.