If the leak is not severe, you may be able to clean the corrosive material from the battery compartment and internal components. Never touch the corrosive material or corroded batteries with bare hands. Always wear protective equipment, including gloves and safety glasses.
You must first discard the old battery. Place it in a plastic bag before disposing of it.
After removing the battery, analyse the spread of the corrosion.
In most cases, the corrosion is limited to the contacts in the battery compartment. This corrosion may keep a new battery from powering the watch.
Neutralize the alkaline leakage using white vinegar or lemon juice. Use a cotton swab dipped in the liquid and gently wipe the corrosion away.
If the corrosion is difficult to remove, you may try using an old toothbrush dipped in white vinegar or lemon juice.
When you apply the vinegar or lemon juice, the alkaline leakage should start to fizz. When the fizzing stops, wipe away the remaining corrosion with a dry cloth.
It is also possible for the corrosion to spread to internal components, requiring you to disassemble the watch to reach the leakage.
If you do not want to risk damaging your watch further, consider taking it to a professional watchmaker or jeweller.
They should have the tools necessary to safely clean the corrosion away.