Hatchets. That is all.

But hatchets or spears of bone would be remarkably brittle as a substance. And as a whole bone tends to soften easier when exposed to excessive amounts of water, which would be in the ocean. Microorganisms would find their way to and eat at the bones far faster than they would on land where you could bury skeletal remains in the sand or a more organic-matter enriched substance like peat-moss in swamps or tar swamps (though the former may still have an effect on the nature of decomposition).

The solution then might be to utilize chunks of hard rocks broken off from undersea mountains or whatever, or perhaps some corals even (which can be viciously sharp). Or as a related-alternative: perhaps utilizing diamonds push up through undersea vents and "farmed" from the porous volcanic rock at the bottom of the sea. Diamonds are naturally hard and you could use one to chip through wood if given time.

Undersea vents may also be a sort of natural forge if you want to go out as far as basic metal-working. Though I would hazard that metals forged underwater may be full of more impurities than metal forged on the surface (due in part probably to the more uncontrollable nature of an undersea vent or volcano in comparison to a land-top forge fueled by coal, charcoal, or simple timber/animal shit). So the strength of a hatchet or spear forged underwater through using volcanic magma would be in question but would be enough to lampshade the issue as a whole.

And if time is also of an issue and keeping pace with a ship without tiring (given I haven't read the post and just reading the arguments) one could latch onto the underside with hooks like with what a butcher or even a logger on the surface would use to grip and manipulate timbers. I imagine mer-folk would need to keep mobile anyways to keep their blood pumped full of oxygen. Unless their mouth breathers.

And despite the thickness of ocean-going hulls they can be easily softened and eaten away by marine life. Straight wooden hulls are subject to damages by worms and barnacles which can compromise the hull of the ship. Every so many years any ship will need to be dragged back onto land to be scraped clean of aquatic wild-life and have their hulls repaired.

In our world the British Empire learned to deal with this through employing Copper Sheathing. The chemical reaction of copper to salt water helped create a chemical barrier that repelled or killed marine organisms and thus greatly extended the ships life at sea as well as lessening the drag on the ship and allowing them to go faster. But this was an expensive thing to use and even develop (they went through a lot of hulls and continually closed and revived the project, the biggest risk they faced being galvanic reaction between the copper and the iron spikes used to hold their ships together which would destroy the copper sheaths). I don't expect this to be really seriously used all things considered, just a fun-fact.