Would it be possible to separate the Co2+ and Ni2+ by precipitation with hydroxide?
No, not with hydroxide. It be possible with (Co2+) SCN- ions.
Co2+ + 4SCN- --> <--- [Co(SCN)4]2- It's chemical proof for Co2+ ion. This complex is blue color.
(Ni2+) you can proof on this way:
Your solution of ions (Co2+ has been proven) in test tube have to be a little acidity. Then you have to add diacethyldioksine (I don't know how exactly is correct.Acetil or actehyl? ) It'll to appear pink colur which is proof for Ni2+ ion.
That's all.