TOWOT Solder Wick Braid 10ft Desoldering Wick Braid Remover Tool With Flux No-Clean Soldering Wick Desolder Soldering 0.1 inch Width

Fortunately, you're trying to open the connection to unlock the timer. Use solder braid to remove the mess and it will probably be fine. Along with a fine tip soldering iron, this is your new best friend: https://www.amazon.com/dp/B0943PDCC4 Next time don't use an axe to work on your watch!