Will mixing methyl 2-cyanoacrylate and cellulose make the cellulose harden quickly?
The working principle of super glue, based on 2-cyano acrylate, is an anionic polymerisation reaction which in principle triggered by any nucleophile.  Water conveniently may be such a nucleophile (the basic anion of water, hydroxyde $\ce{OH^-}$ were even better) which is why the container's mouth originally is not only closed by a cap, but by a thin metal plate.

Cotton consists mainly of cellulose, and cellulose itself



(source)

posesses many hydroxyl groups which like water may trigger the polymerisation.  Since there are so many that are easily accessible, the process is very fast, and proceeds under liberation of much heat (i.e., an exothermic reaction).  Because of this and because the acrylate may generate a strong 3D polymer, one should not wear cloths made of cotton-based fabric, nor gloves of such.

While it is new for me to learn that ionic liquids are able to dissolve cellulose, it might be that such an unmoderated polymerisation "runs away"; i.e. already proceeds at a high rate and, because of the liberated heat and high local concentration of the nucleophiles and monomers, will become even faster, eventually is no longer manageable.  Beside potential splatter of liquid-as-such (thermal threat), the polymerisation reactions could proceed further even on and with the skin of an operator (chemical threat) if not protected properly.  Be warned, as each of the two are potentially dangerous, in addition to the potential hazard that working with ionic liquids, and chemicals in general, represent.

Wikipedia hints to this potentially relevant Critical Review doi 10.1039/C2CS15311D in Chem. Soc. Rev., 2012, 41, 1519-1537.