Identification of aqueous solutions of sodium hydroxide, sodium acetate and potassium thiocyanate using nickel(II)
I know that you would expect a precipitate to form when $\ce{Ni^2+}$ ions are
exposed to $\ce{OH−}$ ions

Yes, you are right, they will indeed form a green precipitate, nickel(II) hydroxide:
$$\ce{NiCl2 + 2NaOH -&gt; Ni(OH)2 + 2NaCl}$$
Nickel(II) hydroxide does not dissolve in excess $\ce{NaOH}$.

I know that you would not expect a precipitate to form when $\ce{Ni^2+}$ ions
are exposed to $\ce{CH3COO−}$ ions

I haven't found any reaction like that to form nickel(II) acetate as it is generally formed from the reaction of nickel(II) carbonate and acetic acid but a little bit of searching gave me a site which shows the formation of nickel(II) acetate from various nickel(II) salts and acetic acid and various salts of acetic acid. The closest reactions I could find from that site is the reaction between acetic acid/lead acetate and nickel(II) chloride and reaction between sodium acetate and nickel(II) nitrate. Technically, the reaction can be written and balanced on pen-paper as it is a simple double displacement reaction but the reaction hasn't been performed practically. Also, I don't know the credibility of the site, so take that site as a reference in your discretion.

I have done some research and concluded that $\ce{NiCl2 + KSCN}$ will not form
a precipitate

Theoretically, it can be formed from a double displacement reaction, but practically it is found to be unfeasible as the end products get contaminated with unreacted reactants and byproducts to the point where yield is somewhat negligible. From a paper1:

Compounds containing nickel(II) thiocyanate are prepared mostly by metathesis between nickel(II) chloride or nitrate with alkali metal
thiocyanates. However, in such syntheses if precautions are not
taken, the final product may be contaminated with starting
materials. Methods known for preparing this compound include:
treatment of a dilute thiocyanic acid solution with nickel hydroxide
or carbonate, followed by evaporating the mixture at $\pu{150 ^\circ C}$; mixing and boiling aqueous solutions of ammonium thiocyanate and
barium hydroxide, followed by addition of an aqueous nickel sulphate.

Conclusion
A: $\ce{NaOH}$
B: $\ce{CH3COONa}$, as it is somewhat a weak base, pH = 8.9
Reference

Sarma, K.P., Poddar, R.K. A convenient method of preparing nickel(II) thiocyanate and its use in synthesis. Transition Met Chem 9, 135–138 (1984). DOI: 10.1007/BF00935928 (PDF)