How can you rekey a gate lock when the gate is locked and you lost the key?

This is generally a job for a locksmith. You can't "cut" the lock on a door or gate to get in and replace it like you can with a padlock or combination lock. It costs a bit of money to have someone come out to your house, but unless you can get the whole mechanism off the gate or door, you'll have to have a pro come out to handle the job. (If you can get to the "back" or "operating" side of the lock and remove it, you can take the whole thing in to a locksmith and get a key made for a lot less than having a technician come out.) Oh, and when you get things squared away, get an extra key or two and have them strategically placed for this kind of emergency. Good luck.