After you have done editing your fleet, save it, then go into the Load Fleet screen and press the button with the Steam logo and the Up arrow:

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Note: If you are updating your fleet, you will receive this confirmation dialog:



Press Update, otherwise Cancel will upload a new fleet. If you press Cancel by mistake, simply go to the new fleet on the Workshop and delete it. Your existing fleet is still at the old link.

This behavior may change it the future when the game is more polished.

Refer to this guide: How to Give Your Workshop Fleets Cool Thumbnails.

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Please let me know if you have any questions in the comments below.