Obviously the extent of damage will be a big factor in the length of time to repair a ship but after a battle using cannons (providing the ships not been sunk) but let's say it was a typical battle. The ships been left with a reasonable amount of damage and has a few holes from the cannon fire.
A few related questions:
Following a naval battle would all the ships that were damaged go into dry dock for repair straight away?
Were crews trained/able to make on the move repairs?
Would they often be required to continue onto the other tasks while still damaged?
There were very large ships that were build for big seabattles, and smaller ones that were build for patrolling, dealing with smugglers etc. Epic sea battles did only occur during wartime, but every seafaring nation had to deal with smugglers and pirates all the time, which would often result in combat on a small scale. So this would be more likely a "typical battle". Combat could mean to hunt down a pirate sloop, enter it, making it a prize and then selling it in a port nearby. It could also mean to send a raiding party of 20 sailors with improvised weapons against a smuggler's nest, with no ships involved except for transportation.
Given a ship that survived a battle against another ship, the fog of cannons and smallarms lifting, the surviving seamen would fight off fire and water by pumping and other measures, then repairing the hull, sails and rigs to get the ship to the next port or friendly shore if possible. Every ship transported spare parts for this, or had to sail to the next shore to acquire enough wood etc. if in short supply.
There was no typical career path in the navy, except for officers. Nowadays a young person can join the navy, learn a profession there that also has a civilian use, and leave after a few years of service - this was not the case back then. Seamen were often workers, skilled or not, or professional craftsmen before they volunteered or were forced into service (the latter was the case for 50% or more of all British seamen in wartime). So when they became seamen, they were separated into men that had no prior profession of value for the navy, these would become marines or sailors, and men that had prior experience in crafts like woodworking, carpenting, caulking, ropemaking, smithing, and tailoring, they would become the ship's carpenters, caulkers, ropemakers, gunsmiths, sailmakers, etc., or would assist in these crafts (= carpenter's mate etc). Sailors and other unskilled workers had to work and sleep in several shifts a day, they were the workforce performing all tasks that require a lot of hands and physical labor, like rigging, raising and lowering the sails and manning the guns, which means they were needed immediately, at any given time of day or night. They were the operators of the ship. The craftsmen, on the other hand, had a normal working day, doing repairs and producing spare parts during daytime, except for emergencies. Wear and tear was a daily problem, so there always was a rope to maintain, a sail to patch, or some planks to seal. A craftsmans's job was not operating the ship, but keeping it from falling apart.
Separating the crew into these two workforces also means that a ship could continue to sail while undergoing most repairs. As long as the ship's core mechanics are working (= it has at least one mast and sails that gives a small amount of speed, a working rudder, and a hull that is intact more or less), its possible to sail. In every other case, the ship would get to the next port or shore for immediate repairs. Warships typically did not operate in high water, but near the coast, because that's were the merchant ships are (=the easy preys or the traffic you want to defend or control). This means that a ship could easily reach a friendly port or shore within a few days. Longer voyages with no port in sight were not common and raised the issue of food preservation, so a navigator would try to find a route with ports or friendly shores nearby for resupply at least once a week.
A captain had to cover expenditures and pay loans on his own account, so he would try not to sink enemy ships, but entering, capturing, and selling it. He could order to disassemble it to repair his own ship, if in shortage of spare parts (like e.g. a broken mast). There were cases in which an entire crew switched to the captured ship, if the previous one was impossible to recover.